Algebraic Number Theory - WordPress.com€¦ · CHAPTER 1. ALGEBRAIC FOUNDATIONS Deﬁnition 1.2.4. An integral domain D that satisﬁes the following three proper-ties: (i) D is

Algebraic Number Theory

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Algebraic Number Theory

Travis Dirle

December 4, 2016

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Contents

1 Algebraic Foundations 11.1 Ring Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Dedekind Domains . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4 Number Fields . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Norms, Traces, and Discriminants 112.1 Norms and Traces . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Discriminants . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3 Integral Bases . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.4 Norms of Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 The Ideal Class Group 173.1 The Ideal Class Group . . . . . . . . . . . . . . . . . . . . . . 173.2 Lattices and Preliminaries . . . . . . . . . . . . . . . . . . . . . 173.3 The Canonical Embedding of a Number Field . . . . . . . . . . 18

4 Units and Dirichlet’s Unit Theorem 214.1 Dirichlet’s Unit Theorem . . . . . . . . . . . . . . . . . . . . . 214.2 Units in Quadratic Fields . . . . . . . . . . . . . . . . . . . . . 22

5 Cyclotomic Fields 255.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2 Integral Basis of a Cyclotomic Field . . . . . . . . . . . . . . . 26

6 Splitting of Primes in Extension Fields 276.1 Lifting of Prime Ideals . . . . . . . . . . . . . . . . . . . . . . 276.2 The Discriminant and Ramification . . . . . . . . . . . . . . . . 296.3 Quadratic Reciprocity . . . . . . . . . . . . . . . . . . . . . . . 29

7 Galois Extensions 337.1 Decomposition and Inertia Groups . . . . . . . . . . . . . . . . 337.2 The Frobenius Automorphism . . . . . . . . . . . . . . . . . . 35

i

CONTENTS

8 The Theory of Valuations 378.1 The p-adic Numbers . . . . . . . . . . . . . . . . . . . . . . . 378.2 The p-adic Absolute Value . . . . . . . . . . . . . . . . . . . . 398.3 Valuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408.4 Completions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438.5 Unramified and Tamely Ramified Extensions . . . . . . . . . . 458.6 Global Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

9 Analytical Methods 519.1 The Classical Zeta-functions . . . . . . . . . . . . . . . . . . . 519.2 Asymptotic Distribution of Ideals and Prime Ideals . . . . . . . 53

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Chapter 1

Algebraic Foundations

1.1 RING EXTENSIONS

Definition 1.1.1. Let S be a commutative ring with identity and R a subring of Scontaining 1S . Then S is said to be an extension ring of R.

Definition 1.1.2. Let S be an extension ring of R and s ∈ S. If there existsa monic polynomial f(x) ∈ R[x] such that s is a root of f, then s is said tobe integral over R. If every element of S is integral over R, S is said to be anintegral extension of R.

Theorem 1.1.3. Let S be an extension ring of R and s ∈ S. Then the followingconditions are equivalent.

(i) s is integral over R;(ii) R[s] is a finitely generated R-module;(iii) there is a subring T of S containing 1S and R[s] which is finitely gener-

ated as an R-module;(iv) there is an R[s]-submodule B of S which is finitely generated as an R-

module and whose annihilator in R[s] is zero.

Corollary 1.1.4. If S is a ring extension of R and S is finitely generated as anR-module, then S is an integral extension of R.

Theorem 1.1.5. If S is an extension ring of R and s1, . . . , st ∈ S are integral overR, then R[s1, . . . , st] is a finitely generated R-module and an integral extensionring of R.

Theorem 1.1.6. If T is an integral extension ring of S and S is an integral exten-sion ring of R, then T is an integral extension ring of R.

Theorem 1.1.7. Let S be an extension ring of R and let R be the set of allelements of S that are integral over R. Then R is an integral extension ring of Rwhich contains every subring of S that is integral over R.

If S is an extension ring of R, then the ring R is called the integral closure ofR in S. If R = R, then R is said to be integrally closed in S. An integral domain

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CHAPTER 1. ALGEBRAIC FOUNDATIONS

R is said to be integrally closed provided R is integrally closed in its quotientfield.

Theorem 1.1.8. Let T be a multiplicative subset of an integral domain R suchthat 0 6∈ T . If R is integrally closed, then T−1R is an integrally closed integraldomain.

If S is an extension ring of R and I(6= S) is an ideal of S, it is easy to seethat I ∩ R 6= R and I ∩ R is an ideal of R. The ideal J = I ∩ R is calledthe contraction of I to R and I is said to lie over J. If Q is a prime ideal in anextension ring S of a ring R, then the contraction Q ∩ R of Q to R is a primeideal of R.

Theorem 1.1.9. (Lying-over Theorem) Let S be an integral extension ring of Rand P a prime ideal of R. Then there exists a prime ideal Q in S which lies overP (that is, Q ∩R = P ).

Theorem 1.1.10. (Going-up Theorem) Let S be an integral extension ring of Rand P1, P prime ideals in R such that P1 ⊂ P . If Q1 is a prime ideal of S lyingover P1, then there exists a prime ideal Q of S such that Q1 ⊂ Q and Q lies overP.

Theorem 1.1.11. Let S be an integral extension ring of R and P a prime idealin R. If Q and Q’ are prime ideals in S such that Q ⊂ Q′ and both Q and Q’ lieover P, then Q = Q′.

Theorem 1.1.12. Let S be an integral extension ring of R and let Q be a primeideal in S which lies over a prime ideal P in R. Then Q is maximal in S if andonly if P is maximal in R.

1.2 DEDEKIND DOMAINS

Definition 1.2.1. (Cohen) A commutative ring R with identity is Noetherian ifand only if every prime ideal of R is finitely generated. Or also, we define aNoetherian ring to be a ring in which every ascending chain of (two-sided)ideals terminates.

Definition 1.2.2. An integral domain D is said to satisfy the maximal conditionif every nonempty set S of ideals of D contains an ideal that is not properlycontained in any other ideal of the set S; that is, S possesses an ideal I such thatif J is an ideal in S with I ⊂ J then J = I .

Theorem 1.2.3. Let D be an integral domain. Then D is Noetherian if and onlyif D satisfies the maximal condition.

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Definition 1.2.4. An integral domain D that satisfies the following three proper-ties:

(i) D is a Noetherian domain,(ii) D is integrally closed, and(iii) each prime ideal of D is a maximal ideal, is called a Dedekind domain.

Theorem 1.2.5. Let K be a number field. Let OK be the ring of integers of K.Then OK is a Dedekind domain.

Theorem 1.2.6. In a Noetherian domain, every nonzero ideal contains a productof one or more prime ideals.

Definition 1.2.7. A is a fractional ideal of D, when A = 1γI , where γ ∈ D\0

and I is an integral ideal of D. This representation is not unique. The fractionalideal is principal when I is principal.

Lemma 1.2.8. If a is an ideal of OK , define

a−1 = α ∈ K : αa ⊂ OK

Then a−1 is a fractional ideal.

Lemma 1.2.9. If a is a proper ideal of OK , then a−1 strictly contains OK .

Theorem 1.2.10. If a is any non-zero of OK , then aa−1 = OK .

Theorem 1.2.11. If D is a Dedekind domain every integral ideal (6= 0, D) is aproduct of prime ideals and this factorization is unique.

Theorem 1.2.12. Let K be a number field. Then every proper integral ideal ofOK can be expressed uniquely up to order as a product of prime ideals.

Theorem 1.2.13. The set of all nonzero integral and fractional ideals of a Dedekinddomain D forms an Abelian group with respect to multiplication. The iden-tity element of the group is (1) = D and the inverse of A =

∏ni=1 P

aii , where

P1, . . . , Pn are distinct prime ideals and a1, . . . , an are integers, is

A−1 =n∏i=1

P−aii .

Theorem 1.2.14. Let K be a number field. Then the set of all nonzero integraland fractional ideals of OK forms an abelian group I(K) with respect to multi-plication.

Definition 1.2.15. With the preceding notation, the order of the nonzero ideal Aof the Dedekind domain D with respect to the prime ideal Pi, written ordPi

(A),is defined by ordPi

(A) = ai. For any prime ideal P 6= P1, . . . , Pn we defineordP (A) = 0.

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Theorem 1.2.16. Let D be a Dedekind domain. Let A and B be nonzero integralor fractional ideals of D. Then

A|B if and only if A ⊃ B.

Theorem 1.2.17. Let D be a Dedekind domain. Let P be a prime ideal of D. LetA and B be nonzero integral or fractional ideals of D. Then

(i) ordP (AB) = ordP (A) + ordP (B),(ii) ordP (A+B) = min(ordP (A), ordP (B)).

Definition 1.2.18. Let D be a Dedekind domain with quotient field K. For α ∈K,α 6= 0, we define the order of α with respect to a prime ideal to be

ordP (α) = ordP (〈α〉)

for any prime ideal P of D.

Proposition 1.2.19. Let D be a Dedekind domain with quotient field K. Let A bea nonzero ideal of D. Let α ∈ K,α 6= 0. Then

α ∈ A if and only if ordP (α) ≥ ordP (A) for all prime ideals P of D.

Theorem 1.2.20. Let D be a Dedekind domain with quotient field K. Given anyfinite set of prime ideals P1, . . . , Pk of D and a corresponding set of integersa1, . . . , ak then there exists α ∈ K such that

ordPi(α) = ai, i = 1, 2, . . . , k,

and ordP (α) ≥ 0, for any prime ideal P 6= P1, . . . , Pk.

Theorem 1.2.21. Let D be a Dedekind domain. Let A be a fractional or integralideal of D. Then A is generated by at most two elements.

Theorem 1.2.22. If R is a Dedekind domain, then R is a unique factorizationdomain if and only if R is a principle ideal domain.

Proposition 1.2.23. Let I be a nonzero ideal of the Dedekind domain R. Thenthere is a nonzero ideal I’ such that II’ is a principal ideal. Moreover, if J is anarbitrary nonzero ideal of R, then I’ can be chosen to be relatively prime to J.

Corollary 1.2.24. A Dedekind domain with only finitely many prime ideals is aprincipal ideal domain.

Theorem 1.2.25. Let I be a nonzero ideal of the Dedekind domain R, and let abe any nonzero element of I. Then I can be generated by two elements, one ofwhich is a.

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1.3 LOCALIZATION

To “localize” means to form quotients, the most familiar case being the pas-sage from an integral domain A to its field of fractions K. More generally,choosing instead of A\0 any nonempty S ⊂ A\0 which is closed undermultiplication. We get a new familiar ring AS−1. The most important specialcase is as follows:

Definition 1.3.1. Take the multiplicative subset to be the complement S = A\pof a prime ideal p of A. We write Ap instead of AS−1 and call the ring thelocalization of A at p

Proposition 1.3.2. The mappings

q 7→ qS−1 and D 7→ D ∩ A

are mutually inverse 1− 1 correspondences between the prime ideals q ⊂ A\Sof A and the prime ideals D of AS−1.

Usually S will be the complement of a union⋃

p∈X p over a setX of prime idealsof A, in this case one writes

A(X) = f/g : f, g ∈ A, g 6≡ 0 mod p for p ∈ X

In the case that X consists of only one prime ideal p, we have

Ap = f/g : f, g ∈ A, g 6≡ 0 mod p.

Corollary 1.3.3. If p is a prime ideal of A, then Ap is a local ring, i.e., it has aunique maximal ideal, namely mp = pAp. There is a canonical embedding

A/p→ Ap/mp,

identifying Ap/mp with the field of fractions of A/p. In particular, if p is amaximal ideal of A, then one has

A/pn ∼= Ap/mnp for n ≥ 1.

In a local ring with maximal ideal m, every element a /∈ m is a unit. The sim-plest local rings, except for fields, are the following:

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Definition 1.3.4. A discrete valuation ring is a principal ideal domain o with aunique maximal ideal p 6= 0.

The maximal ideal is of the form p = (π) = πo, for some prime element π.Since every element not contained in p is a unit, it follows that, up to associatedelem, π is the only prime elem of o. Every nonzero elem of o may therefore bewritten as επn, for some ε ∈ o∗, and n ≥ 0. More generally, every element a 6= 0of the field of fractions K may be uniquely written as

a = επn, for ε ∈ o∗ and n ∈ Z.

Definition 1.3.5. The exponent n is called the valuation of a. It is denoted v(a),and is characterized by the equation

(a) = pv(a).

The valuation is a function v : K∗ → Z, and extending it to K is done by lettingv(0) =∞.

The discrete valuation rings arise as localizations of Dedekind domains. For aDedekind domain o, we have for each prime ideal p 6= 0 the DVR op and thecorresponding valuation, sometimes called the exponential valuation

vp : K∗ → Z

of the field of fractions. The significance of these valuations lies in their relationto the prime ideal factorization. If x ∈ K∗ and (x) =

∏pvp is the prime fact. of

(x), then for each p, one has vp = vp(x).

Definition 1.3.6. The localization of Z at the prime ideal (p) = pZ is given by

Z(p) = a/b : a, b ∈ Z, p - b.

The maximal ideal pZ(p) consists of all fractions a/b satisfying p | a, p - b, andthe group of units consists of all fractions a/b satisfying p - ab. The valuationassociated to Z(p),

vp : Q→ Z ∪∞is called the p-adic valuation of Q. For x ∈ Q∗ it is given by vp(x) = v, wherex = pva/b with integers a, b relatively prime to p.

Definition 1.3.7. Let o be the normalization of o, i.e., the integral closure of oin K. A prime p 6= 0 of o is called regular if op is integrally closed, and thus aDVR. The conductor of o is the biggest ideal of o which is contained in o, i.e.,f = a ∈ o : ao ⊂ o. There are only finitely many non-regular prime ideals ofo, namely the divisors of the conductor of o.

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Proposition 1.3.8. For any prime ideal p 6= 0 of o one has

p - f ⇔ p is regular.

If this is the case, then p = po is a prime ideal of o and op = op.

1.4 NUMBER FIELDS

Definition 1.4.1. A complex number is an algebraic integer if and only if it isa root of some monic polynomial with coefficients in Z. It is a complex numberwhich is integral over Z.

Definition 1.4.2. A complex number that is algebraic over Q is called an alge-braic number.

Definition 1.4.3. A number field is a finite extension of Q. It is a subfield of Cof the form Q(α1, . . . , αn), where α1, . . . , αn are algebraic numbers.

Theorem 1.4.4. If K is an algebraic number field then there exists an algebraicnumber θ such that K = Q(θ).

Theorem 1.4.5. If K is an algebraic number field then there exists an algebraicinteger θ such that K = Q(θ).

Definition 1.4.6. The set of all algebraic integers that lie in the number field Kis denoted OK; if Ω is the domain of all algebraic integers, then OK = Ω ∩K.OK is called the ring of integers of the number field K.

Theorem 1.4.7. Let D be a unique factorization domain. Let F be the field ofquotients of D. Then c ∈ F is integral over D if and only if c ∈ D.

Theorem 1.4.8. Every algebraic number is of the form a/b, where a is an alge-braic integer and b is a nonzero ordinary integer.

Theorem 1.4.9. If K is a number field then the quotient field of OK is K.

Theorem 1.4.10. If K is a number field then OK is integrally closed.

Theorem 1.4.11. Let K be a number field. Then OK is a Dedekind domain.

Theorem 1.4.12. Let K be a number field. Then every nonzero ideal in OKcontains a nonzero rational integer.

Theorem 1.4.13. Let K be a number field. Let I be a nonzero ideal of OK . Thenthere exists γ ∈ I such that K = Q(γ).

Theorem 1.4.14. Let K be a number field of degree n over Q. Then there areexactly n distinct monomorphisms σk : K → C (k = 1, . . . , n).

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Definition 1.4.15. Let K be a number field. Let θ be an algebraic number suchthat K = Q(θ). Let θ1 = θ, θ2, . . . , θn be the conjugates of θ over Q. Then thefields

Q(θ1) = Q(θ) = K,Q(θ2), . . . ,Q(θn)

are called the conjugate fields of K.

Theorem 1.4.16. Let K be a number field. Let θ be an algebraic number suchthat K = Q(θ). Let θ1 = θ, θ2, . . . , θn be the conjugates of θ. Let φ be anotheralgebraic number such that K = Q(φ). Let c0, c1, . . . cn−1 ∈ Q be such that fork = 1, 2, . . . , n

φk = c0 + c1θk + · · ·+ cn−1θn−1k

Then φ1, φ2, . . . , φn are the conjugates of φ over Q, and

Q(θk) = Q(φk), k = 1, 2, . . . , n.

Definition 1.4.17. Let K be a number field of degree n over Q. Let θ ∈ K besuch that K = Q(θ). Let θ1 = θ, θ2, . . . , θn be the conjugates of θ. For α ∈ Kthere exist unique rational numbers c0, c1, . . . , cn−1 such that α = c0+c1θ+· · ·+cn−1θ

n−1. For k = 1, 2, . . . , n we set αk = c0 + c1θk + · · ·+ cn−1θn−1k ∈ Q(θk).

The set of algebraic numbers α1 = α, α2, . . . , αn is called a K-conjugates ofα.

The conjugates of α relative to K are obtained from α by applying the monomor-phisms σk : K → C to α. It can be shown that the conjugates of α relative to Kdo not depend on the choice of θ such that K = Q(θ).

Definition 1.4.18. Let K be a number field of degree n. Let α ∈ K. Let α1 =α, α2, . . . , αn be the K-conjugates of α. Then the field polynomial of α over Kis the polynomial

fldK(α) =n∏k=1

(x− αk).

Theorem 1.4.19. Let K be a number field of degree n. Let α ∈ K. Then

fldK(α) ∈ Q[x].


fldK(α) = (irrQ(α))s

where s is the positive integer

s =n

deg(irrQ(α));

Theorem 1.4.21. Let K be a number field. Let α ∈ OK . Then the K-conjugatesof α are algebraic integers.

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Theorem 1.4.22. Let K be a number field. Let α ∈ K. Then all the K-conjugatesof α are equal if and only if α ∈ Q.

Theorem 1.4.23. Let K be a number field. Let α ∈ K. Then all the K-conjugatesof α are distinct if and only if K = Q(α).

Proposition 1.4.24. The set B of algebraic integers of Q(√m), m square-free,

can be described as follows:(i) If m 6≡ 1 mod 4, then B consists of all a+ b

√m, a, b ∈ Z;

(ii) If m ≡ 1 mod 4, then B consists of all (u/2) + (v/2)√m,u, v ∈ Z,

where u and v have the same parity.

Note that since m is square-free, then it is not divisible by 4, so condition (i)can be written as m ≡ 2 or 3 mod 4.

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10

Chapter 2

Norms, Traces, and Discriminants

2.1 NORMS AND TRACES

Definition 2.1.1. LetE/F be a separable extension of degree n, and let σ1, . . . , σnbe the distinct F-monomorphisms of E into an algebraic closure of E. Then wedefine the norm, trace and the characteristic polynomial as

NE/F (x) =n∏i=1

σi(x)

TE/F (x) =n∑i=1

σi(x)

charE/F (x) =n∏i=1

(X − σi(x)).

Proposition 2.1.2. We have that charE/F (x) = [min(x, F )]r, where r = [E :F (x)].

Corollary 2.1.3. Let [E : F ] = n and [F (x) : F ] = d. Let x1, . . . , xd be theroots of min(x, F ), counting multiplicity, in a splitting field. Then

N(x) = (d∏i=1

xi)n/d

T (x) =n

d

d∑i=1

xi

char(x) = [d∏i=1

(X − xi)]n/d.

Proposition 2.1.4. If F ⊂ K ⊂ E, where E/F is finite and separable, then

TE/F = TK/F TE/K and NE/F = NK/F NE/K .

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CHAPTER 2. NORMS, TRACES, AND DISCRIMINANTS

Definition 2.1.5. IfE/F is a field extension of degree n and x ∈ E, then we havethe F-linear transformation m(x) given by multiplication by x, that is, m(x)y =xy. If the matrix A(x) = [aij(x)] represents m(x) with respect to some basis forE over F, then the norm and trace of x is given by

NE/F (x) = |A(x)| and TE/F (x) = Tr(A(x))

And the characteristic polynomial of x is defined as the characteristic polyno-mial of the matrix A(x), that is,

charE/F (x) = |(XI − A(x))|

The norm, trace, and coefficients of the characteristic polynomial are elementsbelonging to the base field F.

We have the following properties:

char(x) = Xn − T (x)Xn−1 + · · ·+ (−1)nN(x)

If x, y ∈ E and a, b ∈ F , then

T (ax+ by) = aT (x) + bT (y) and N(xy) = N(x)N(y).

If a ∈ F , then

N(a) = an, T (a) = na, and char(a) = (X − a)n.

Proposition 2.1.6. IfE/F is a finite separable extension, then the trace TE/F (x) 6=0 for every x ∈ E.

Proposition 2.1.7. Let A be an integral domain, K its field of fractions, L anextension of finite degree of K, and x an element of L integral over A. Assumechar K = 0. Then the coefficients of the characteristic polynomial of x relativeto L and K, in particular TrL/K(x) and NL/K(x), are integral over A.

Corollary 2.1.8. Suppose further, that A is integrally closed. Then the coeffi-cients of the characteristic polynomial of x, in particular the trace and norm ofx, are actually elements of A.

Corollary 2.1.9. An algebraic integer a that belongs to Q must in fact belong toZ.

2.2 DISCRIMINANTS

Definition 2.2.1. Let K be a number field of degree n. Let ω1, . . . , ωn be n ele-ments of K. Let σk (k = 1, 2, . . . , n) denote the n distinct monomorphisms: K →

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C. For i = 1, . . . , n let ω(1)i = σ1(ωi) = ωi, ω

(2)i = σ2(ωi), . . . , ω

(n)i = σn(ωi)

denote the conjugates of ωi relative to K. Then the discriminant of ω1, . . . , ωnis

D(ω1, . . . , ωn) =

∣∣∣∣∣∣∣∣∣ω

(1)1 ω

(1)2 · · · ω

(1)n

ω(2)1 ω

(2)2 · · · ω

(2)n

...... · · · ...

ω(n)1 ω

(n)2 · · · ω

(n)n

∣∣∣∣∣∣∣∣∣

2

Alternatively written: for ω = (ω1, . . . , ωn) then D(ω) = [det(σi(ωj))]2.

Definition 2.2.2. If [L : K] = n, then the discriminant of the n-tuple x =(x1, . . . , xn) of elements of L is

D(x) = |TL/K(xixj)|.

Thus we form a matrix whose ij entry is the trace of xixj , and take the determi-nant.

Definition 2.2.3. Let K be a number field of degree n. Let α ∈ K. Then wedefine the discriminant D(α) of α by

D(α) = D(1, α, α2, . . . , αn−1).


D(α) =∏

1≤i<j≤n

(α(i) − α(j))2,

where α(1) = α, α(2), . . . , α(n) are the conjugates of α with respect to K.

Definition 2.2.5. Let f(x) = anxn + an−1x

n−1 + · · · + a1x + a0 ∈ C[x]. Letx1, . . . , xn ∈ C be the roots of f(x). The discriminant of the polynomial f(x)is

disc(f(x)) = a2n−2n

∏1≤i<j≤n

(xi − xj)2 ∈ C


D(α) = disc(fldK(α)).


K = Q(α) if and only if D(α) 6= 0.

Theorem 2.2.8. Let K be a number field of degree n.(i) If x1, . . . , xn ∈ K then D(x1, . . . , xn) ∈ Q.(ii) If x1, . . . , xn ∈ OK then D(x1, . . . , xn) ∈ Z.(iii) If x1, . . . , xn ∈ K then D(x1, . . . , xn) 6= 0 if and only if x1, . . . , xn are

linearly independent over Q.

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Definition 2.2.9. Let K be a number field of degree n. Let I be a nonzero idealof OK . Let x1, . . . , xn be a basis of I. Then the discriminant of the ideal I isthe nonzero integer given by

D(I) = D(x1, . . . , xn).

2.3 INTEGRAL BASES

Definition 2.3.1. Let K be a number field. A basis for OK is called an integralbasis for K.

Theorem 2.3.2. Let K be a quadratic field. Let m be the unique squarefreeinteger such that K = Q(

√m). Then 1,

√m is an integral basis for K if

m 6≡ 1 mod 4 and 1, 1+√m

2 is an integral basis for K if m ≡ 1 mod 4.

Definition 2.3.3. Let K be a number field of degree n. Let x1, . . . , xn be an in-tegral basis for K. Then the field discriminant, denoted d(K) is D(x1, . . . , xn).

Theorem 2.3.4. Let K be a quadratic field. Let m be the unique squarefreeinteger such that K = Q(

√m). Then the discriminant d(K) of K is given by

d(K) =

4m, if m 6≡ 1 mod 4

m, if m ≡ 1 mod 4

Theorem 2.3.5. Let K be a quadratic field. Then K = Q(√d(K)).

Note that the quadratic field K is a real field if and only if d(K) > 0.

Theorem 2.3.6. Let K be a number field. Then d(K) ≡ 0 or 1 mod 4.

Theorem 2.3.7. Let K be a number field of degree n. Let x1, . . . , xn ∈ OK besuch that D(x1, . . . , xn) 6= 0. Then

|d(K)| ≤ |D(x1, . . . , xn)|.

Moreover, if D(x1, . . . , xn) is squarefree then x1, . . . , xn is an integral basisfor OK .

Theorem 2.3.8. Let K be a number field of degree n. Let x1, . . . , xn ∈ OK besuch that D(x1, . . . , xn) 6= 0. Then for each α ∈ OK there exist unique rationalintegers y1, . . . , yn such that

α =n∑j=1

yjD(x1, . . . , xn)

xj

and D(x1, . . . , xn)|y2j , j = 1, 2, . . . , n.

14


Theorem 2.3.9. Let K be a number field of degree n. Let θ ∈ OK be such thatK = Q(θ). Let α ∈ OK . Then there exist unique rational numbers rj/sj(j =1, 2, . . . , n) with (rj, sj) = 1 and sj > 0 such that

α =n∑j=1

rjsjθj−1

and 1 ≤ sj ≤ |D(θ)|, s2j |D(θ).

2.4 NORMS OF IDEALS

Definition 2.4.1. Let K be a number field of degree n. Let I be a nonzero ideal ofOK . Then the norm of the ideal I, written N(I), is the positive integer definedby

N(I) =

√D(I)

d(K)

Proposition 2.4.2. Let K be a number field. If x is a nonzero element of OK ,then |NK/Q(x)| = card(OK/xOK).

Definition 2.4.3. Given a nonzero integral ideal I of OK , we call the number|OK/I| the norm of I and denote it by N(I).

Proposition 2.4.4. Let b be any nonzero element of the ideal I of OK , and letm = NK/Q(b) ∈ Z. Then m ∈ I and |OK/mOK | = mn, where n = [K : Q].

Corollary 2.4.5. If I is any nonzero ideal of OK , then N(I) is finite. In fact, ifm is as before, then N(I)|mn.

Proposition 2.4.6. If I = (a) with a 6= 0, then N(I) = |NK/Q(a)|.

Theorem 2.4.7. If I and J are nonzero ideals ofOK , then N(IJ) = N(I)N(J).

Corollary 2.4.8. Let I be a nonzero ideal of OK . If N(I) is prime, then I is aprime ideal.

Proposition 2.4.9. N(I) ∈ I , in other words, I divides N(I). More precisely, Idivides the principal ideal generated by N(I).

Corollary 2.4.10. If I is a nonzero prime ideal of OK , then I divides (contains)exactly one rational prime p.

If we can compute the norm of every nonzero prime ideal P, then by mul-tiplicativity, we can calculate the norm of any nonzero ideal. Let p be the

15


unique rational prime in P, and recall that the relative degree of P over p isf(P ) = [OK/P : Z/pZ]. Therefore

N(P ) = |OK/P | = pf(P )

Proposition 2.4.11. Let A ⊂ B and K ⊂ L be as usual. Theni) For any nonzero ideal a ⊂ A,NL/K(aB) = am, where m = [L : K].ii) Suppose L is Galois over K. Let P be a nonzero prime ideal of B and let

p = P ∩ A. Write pB = (P1 · · ·Pg)e. Then

N(PB) = (P1 · · ·Pg)ef = Πσ∈Gal(L/K)σP.

iii) For any nonzero element b ∈ B,N(b)A = N(bB).

Proposition 2.4.12. A rational integer m can belong to only finitely many idealsof OK .

Corollary 2.4.13. Only finitely many ideals can have a given norm.

16

Chapter 3

The Ideal Class Group

3.1 THE IDEAL CLASS GROUP

Definition 3.1.1. Let K be a number field. Let I(K) be the group of nonzerofractional and integral ideals of OK . Let P (K) be the subgroup of principalideals of I(K). Then the factor group I(K)/P (K) is called the ideal classgroup of K and is denoted by H(K).

Definition 3.1.2. Let K be a number field. The order of the ideal class groupH(K) is called the class number of K and is denoted by h(K).

Notice that if HK is the trivial group, then every ideal is principal, whichimplies unique factorization. The number hK measures the proportion of idealswhich are principal, that is, 1/hK is the proportion of principal ideals. Also,since hK is finite, then for any ideal a we have ahK is principal.

3.2 LATTICES AND PRELIMINARIES

Definition 3.2.1. Let Zn be the subset of Rn given by

Zn = (x1, . . . , xn) ∈ Rn : x1, . . . , xn ∈ Z.

The elements of Zn are called lattice points and Zn is called a lattice. Moregenerally, a discrete subgroup of rank n of Rn is called a lattice in Rn.

Letting e1, . . . , en ∈ Rn, with the ei linearly independent over R, then the eiform a basis for Rn as a vector space over R. The ei also form a basis for a freeZ-module of rank n, namely

H = Ze1 + · · ·+ Zen.

A set constructed this way is a latice in Rn.In the most familiar case, e1 and e2 are linearly independent vectors in the

plane, and T is the parallelogram generated by the ei. In general, every point of

17

CHAPTER 3. THE IDEAL CLASS GROUP

Rn is congruent modulo H to a unique point of T, so Rn is the disjoint union ofsets h+ T, h ∈ H .

Definition 3.2.2. The half open parallelotope called the fundamental domainof H is given by

T = x ∈ Rn : x =n∑i=1

aiei, 0 ≤ ai < 1

Definition 3.2.3. If µ is Lebesgue measure, then the volume µ(T ) of the funde-mental domain T will be denoted v(H).

The word ’volume’ here is an abuse of language since µ(H) = 0.

Lemma 3.2.4. Let S be a Lebesgue measurable subset of Rn with µ(S) > v(H).Then there exist dinstinct points x, y ∈ S such that x− y ∈ H .

Theorem 3.2.5. (Minkowski’s Convex Body Theorem) Let H be a lattic in Rn,and assume that S is a Lebesgue measurable subset of Rn that is symmetric aboutthe origin and convex. If

(i) µ(S) > 2nv(H), or(ii) µ(S) ≥ 2nv(H) and S is compact,then S ∩ (H\0) 6= ∅.

We will use n-dimensional integration techniques, which is needed in theproof that the ideal class group is finite. We will work in Rn, realized as theproduct of r1 copies of R and r2 copies of C, where r1 + 2r2 = n. Out interestis the set

Bt = (y1, . . . , yr1 , z1, . . . , zr2) ∈ Rr1 × Cr2 :

r1∑i=1

|yi|+ 2

r2∑j=1

|zj| ≤ t, t ≥ 0.

We end up having that the volume of Bt is given by

V (r1, r2, t) = 2r1(π

2

)r2 tnn!.

3.3 THE CANONICAL EMBEDDING OF A NUMBER FIELD

Definition 3.3.1. Let L be a number field of degree n over Q, and let σ1, . . . , σnbe the Q-monomorphisms of L into C. If σi maps entirely into R, we say that σiis a real embedding; otherwise it is a complex embedding.

Since the complex conjugate of a Q-monomorphism is also a Q-monomorphism,we can renumber the σi so that the real embeddings are σ1, . . . , σr1 and thecomplex embeddings are σr1+1, . . . , σn with σr1+j paired with its complex con-jugate σr1+r2+j, j = 1, . . . , r2. Thus there are 2r2 complex embeddings, andr1 + 2r2 = n.

18


Definition 3.3.2. The canonical embedding σ : L → Rr1 × Cr2 = Rn is theinjective ring homomorphism given by

σ(x) = (σ1(x), . . . , σr1+r2(x)).

Now for some matrix manipulations. Let x1, . . . , xn ∈ L be linearly indepen-dent over Z (hence the xi form a basis for L over Q). Let C be the matrix whosekth column (k = 1, . . . , n) is

σ1(xk), . . . , σr1(xk),<σr1+1(xk),=σr1+1(xk), . . . ,<σr1+r2(xk),=σr1+r2(xk).

The determinant of C looks something like a discriminant, we have that

detC = (2i)−r2 det(σj(xk))

Proposition 3.3.3. If M is the free Z-module generated by the xi, so that σ(M)is a free Z-module with basis σ(xi), i = 1, . . . , n, then σ(M) is a lattice inRn. The fundamental domain is a parallelotope whose sides are the σ(xi), andthe volume of the fundamental domain is the absolute value of the determinantwhose rows (or colomns) are the σ(xi). Hence

v(σ(M)) = | detC| = 2−r2| detσj(xk)|.

Proposition 3.3.4. Let I be a nonzero integral ideal of OL, so that σ(I) is alattice in Rn. Then the volume of the fundamental domain of this lattice is

v(σ(I)) = 2−r2|d|1/2N(I);

in particular, v(σ(OL)) = 2−r2|d|1/2, where d is the field discriminant.

Theorem 3.3.5. (Minkowski Bound on Element Norm) If I is a nonzero integralideal of OL, then I contains a nonzero element x such that

|NL/Q(x)| ≤(

4

π

)r2 ( n!

nn

)|d|1/2N(I).

Theorem 3.3.6. (Minkowski Bound on Ideal Norm) Every ideal class of L con-tains an integral ideal I such that

N(I) ≤(

4

π

)r2 ( n!

nn

)|d|1/2.

Definition 3.3.7. The number on the right side of the previous theorem is calledthe Minkowski bound. The term CL = n!

nn

(4π

)r2 is called the Minkowski con-stant.

Theorem 3.3.8. In C there are finitely many number fields with a given discrim-inant d.

Theorem 3.3.9. For L a number field, and k a positive integer, there are onlyfinitely many integral ideals I of OL with N(I) = k.

Theorem 3.3.10. The ideal class group of a number field is finite.

19


20

Chapter 4

Units and Dirichlet’s Unit Theorem

4.1 DIRICHLET’S UNIT THEOREM

Definition 4.1.1. For K a number field, the group of units of K, denoted UK ,is the multiplicative group of invertible elements of OK . The group of roots ofunity in K is denoted uK .

Lemma 4.1.2. For L a number field, let UL be the group of units of OL. Anelement x ∈ OL belongs to UL if and only if N(x) = ±1.

Definition 4.1.3. Let σ : L → Rr1 × Cr2 = Rn be the canonical embedding.The logarithmic embedding is the mapping λ : L∗ → Rr1+r2 given by

λ(x) = (log |σ1(x)|, . . . , log |σr1+r2(x)|)Since the σi are monomorphisms, λ(xy) = λ(x)+λ(y), so λ is a homomorphism.

Corollary 4.1.4. The kernel G of the homomorphism λ restricted to UL is a finitegroup.

Proposition 4.1.5. Let H be a finite subgroup of K∗ where K is an arbitraryfield. Then H consists of roots of unity and is cyclic.

Proposition 4.1.6. The group G (our kernel) consists of all the roots of unity inthe field L.

Proposition 4.1.7. UL is a finitely generated abelian group, isomorphic to G×Zs where s ≤ r1 + r2. Dirichlet’s unit theorem says that s actually equalsr1 + r2 − 1.

Theorem 4.1.8. (Dirichlet’s Unit Theorem) For a number field L of degree n, wehave that UL ∼= G × Zs, where G is a finite cyclic group consisting of all theroots of unity in L, and s = r1 + r2 − 1.

Corollary 4.1.9. Assuming prior notation, then there are r = r1 + r2 − 1 unitsu1, . . . , ur in OL such that every unit of OL can be expressed uniquely as

u = zun11 · · ·unr

r

where the ui are algebraic integers and z is a root of unity in L.

21

CHAPTER 4. UNITS AND DIRICHLET’S UNIT THEOREM

Definition 4.1.10. We call u1, . . . , ur a fundamental system of units for thenumber field L.

Definition 4.1.11. Let K be a number field. Let u1, . . . , uk be units of OK . Theunits are said to be independent if and only if

ur11 · · ·urkk = 1⇒ r1 = · · · = rk = 0.

Definition 4.1.12. For any system u1, . . . , ur of units of K (with r = r1 + r2− 1)we define the regulator R(u1, . . . , ur) as the absolute value of the determinantof the matrix [log vj(ui)], where vj runs over all valuations from S∞ except one.

Definition 4.1.13. Let ε1, . . . , εr denote a set of fundamental units, where r =r1 + r2 − 1. Consider the map λ : K → Rr1+r2 above, and write λ(x) =(λ1(x), . . . , λr1+r2(x)), so that λi(x) = log |ρi(x)| if 1 ≤ i ≤ r1 and is log |σi−r1(x)|2if i > r1. Consider the (r + 1) × r-matrix whose entries are λi(εj), and definethe regulator RK to be the absolute value of the determinant of any r× r-minorof this matrix.

Definition 4.1.14. Let t = r+s−1, and let u1, . . . , ut be a system of fundamentalunits. The vector L(ui) ∈ Rr+s projects to

`(ui) = (log |σ1ui|, . . . , log |σrui|, 2 log |σr+1ui|, . . . , 2 log |σtui|)

in Rt, and the vectors `(ui) generate a lattice `(U) in Rt. The regulator of Kis defined to be the determinant of the matrix whose ith row is `(ui). Thus, up tosign, the regulator is the volume of a fundamental domain for `(U).

Corollary 4.1.15. (i) The value of the regulator does not depend on the deletedvaluation.

(ii) The regulator R(u1, . . . , ur) vanishes if and only if the units ui are multi-plicatively dependent.

(iii) If u1, . . . , ur and s1, . . . , sr are two fundamental systems of units of K,then

R(u1, . . . , ur) = R(s1, . . . , sr).

If we denote this common value by R(K), and a1, . . . , ar is an arbitrary systemof multiplicatively independent units of K, then

R(K) ≤ R(a1, . . . , ar).

4.2 UNITS IN QUADRATIC FIELDS

Imaginary Quadratic Fields:

22


First we look at number fieldsL = Q(√m), where m is a square-free negative

integer. There are no real embeddings, so r1 = 0 and 2r2 = n = 2, hence r2 = 1.But then r1 + r2 − 1 = 0, so the only units in OL are the roots of unity in L.

Case 1: Assumem 6≡ 1 mod 4. So an algebraic integer has the form x = a+b√m for integers a and b. Also x is a unit if and only if N(x) = a2−mb2 = ±1.

Thus if m ≤ −2, then b = 0 and a = ±1. If m = −1, we have the additionalpossibility a = 0, b = ±1.

Case 2: Assume m ≡ 1 mod 4. So x = a + b(1 +√m)/2, and N(x) =

(a + b/2)2 − mb2/4 = [(2a + b)2 − mb2]/4. Thus x is a unit if and only if(2a + b)2 −mb2 = 4. So if m ≤ −7, then b = 0, a = ±1. If m = −3, we havethe additional possibilities: a = 0, b = ±1; a = 1, b = −1; a = −1, b = 1.

To summarize, ifOL is the ring of algebraic integers of an imaginary quadraticfield, then the group G of units of OL is 1,−1, except in the following twocases

(i) If L = Q(i), then G = 1, i,−1,−i, the group of 4th roots of unity in L.(ii) If L = Q(

√−3), then G = [(1 +

√−3)/2]j, j = 0, . . . , 5, the group of

6th roots of unity in L.

Theorem 4.2.1. Let K = Q(√d), with d negative and squarefree. Then λ is a

unit in OK if and only if λ is a root of unity, and the units in OK are;

U(OK) =

µ4 = ±1,±i, if d = −1,

µ6 = ±1,±ω,±ω2, if d = −3,

µ2 = ±1, otherwise .

Real Quadratic Fields:Now we examine L = Q(

√m), where m is a square-free positive integer.

Since the Q-automorphisms of L are the identity and a+b√m 7→ a−b

√m, there

are two real embeddings and no complex embeddings. Thus r1 = 2, r2 = 0, andr1 + r2 − 1 = 1. The only roots of unity in R are ±1, so the group of units inthe ring of algebraic integers is isomorphic to −1, 1 × Z. If u is a unit and0 < u < 1, then 1/u is a unit and 1/u > 1. Thus the units greater than 1 are hn,where h is the unique generator greater than 1, called the fundamental unit ofL.

Case 1: Assume m 6≡ 1 mod 4. If x = a + b√m is a solution to N(x) =

a2 − mb2 = ±1, then the four numbers ±a ± b√m are x,−x, x−1,−x−1 in

some order. The fundamental unit, which is the smallest unit greater than 1 canbe found as follows. Compute mb2 for b = 1, 2, . . . , and stop at the first numbermb2

1 that differs from a square a21 by ±1. Then a1 + b1

√m is the fundamental

unit. Continued fraction expansion of√m is more efficient.

Case 2: Assume m ≡ 1 mod 4. Then the algebraic integers are of the formx = 1

2(a+ b

√m), where a and b are integers of the same parity. Since the norm

of x is 14(a2 −mb2), x is a unit if and only if a2 −mb2 = ±4. To calculate the

23


fundamental unit, compute mb2, b = 1, 2, . . . , and stop at the first number mb21

that differs from a square a21 by ±4. The fundamental unit is 1

2(a1 + b1

√m).

Proposition 4.2.2. The positive units of a real quadratic field K ⊂ R form amultiplicative group isomorphic to Z.

Definition 4.2.3. This group contains one and only one generator larger thanone, we call it the fundamental unit.

Theorem 4.2.4. Let K be a number field. Then every unit in OK is a root ofunity if and only if K = Q or K is an imaginary quadratic field.

Theorem 4.2.5. Let K be a number field. Then K possesses a fundamental unitif and only if K is a real quadratic field, a cubic field with exactly one realembedding, or a totally imaginary quartic field.

Theorem 4.2.6. Let K be a number field. Then OK contains only finitely manyroots of unity.

Theorem 4.2.7. Let K be a number field of odd degree n. Then the only roots ofunity in OK are ±1.

Theorem 4.2.8. The only roots of unity in the ring of integers of a cubic field are±1.

24

Chapter 5

Cyclotomic Fields

5.1 PRELIMINARIES

Recall that a cyclotomic extension Q(ζn) is formed by adjoining a primitive nthroot of unity ζn.

Definition 5.1.1. The cyclotomic polynomial Φn(X) is defined as the productof the terms X − ζ , where ζ ranges over all the primitive nth roots of unity in C.Now, an nth root of unity is a primitive dth root of unity for some divisor d of n,so Xn− 1 is the product of all cyclotomic polynomials Φd(X). In particular, letn = pr be a prime power. Since a divisor of pr is either pr or a divisor of pr−1,we have

Φpr(X) =Xpr − 1

Xpr−1 − 1=tp − 1

t− 1= 1 + t+ · · ·+ tp−1

where t = Xpr−1. If X = 1 then t = 1, and it follows that Φpr(1) = p.

Lemma 5.1.2. Let ζ and ζ ′ be primitive (pr)th roots of unity. Then u = (1 −ζ ′)/(1− ζ) is a unit in Z[ζ], hence in the ring of algebraic integers.

Lemma 5.1.3. Let π = 1− ζ and e = φ(pr) = pr−1(p−1), where φ is the Eulerphi function. Then the principal ideals (p) and (π)e coincide.

Proposition 5.1.4. The degree of the extension Q(ζ)/Q equals the degree of thecyclotomic polynomial, namely φ(pr). Therefore the cyclotomic polynomial isirreducible over Q.

Lemma 5.1.5. We have that (π) is a prime ideal (equivalently, π is a primeelement) of OQ(ζ). The relative degree f of (π) over (p) is 1, hence the injectionZ/(p)→ OQ(ζ)/(π) is an isomorphism.

Proposition 5.1.6. N(1−ζ) = ±p, and more generally,N(1−ζps) = ±pps , 0 ≤s < r.

Proposition 5.1.7. Let D be the discriminant of the basis 1, ζ, . . . , ζφ(pr)−1. ThenD = ±pc, where c = pr−1(pr − r − 1).

25

CHAPTER 5. CYCLOTOMIC FIELDS

Lemma 5.1.8. For every positive m, we have Z[ζ] + pmOQ(ζ) = OQ(ζ).

Theorem 5.1.9. The set 1, ζ, . . . , ζφ(pr)−1 is an integral basis for the ring ofintegers of Q(ζpr).

Corollary 5.1.10. A prime number p is ramified in Q(ζn) if and only if n ≡ 0mod p, except in the case where p = 2 = (4, n). A prime number p 6= 2 istotally split in Q(ζn) if and only if p ≡ 1 mod n.

5.2 INTEGRAL BASIS OF A CYCLOTOMIC FIELD

In this section, let K and L be number fields of respective degrees m and n overQ.

Lemma 5.2.1. Assume that [KL : Q] = mn. Let σ be an embedding of K inC and τ an embedding of L in C. Then there is an embedding of KL in C thatrestricts to σ on K and to τ on L.

Lemma 5.2.2. Again assume [KL : Q] = mn. Let a1, . . . , am and b1, . . . , bn beintegral bases for OK and OL respectively. If α ∈ OKL, then

α =m∑i=1

n∑j=1

cijraibj, cij ∈ Z, r ∈ Z

with r having no factor (except ±1) in common with all the cij .

Proposition 5.2.3. Again assume [KL : Q] = mn. If d is the greatest commondivisor of the discriminant of OK and the discriminant of OL, then OKL ⊂1dOKOL. Thus if d = 1, then OKL = OKOL.

Lemma 5.2.4. Let ζ be a primitive nth root of unity, and denote the discriminantof 1, ζ, . . . , ζφ(n)−1 by disc(ζ). Then disc(ζ) divides nφ(n).

Theorem 5.2.5. If ζ is a primitive nth root of unity, then the ring of algebraicintegers of Q(ζ) is Z[ζ]. In other words, the powers of ζ form an integral basis.

Theorem 5.2.6. In general, the field discriminant of Q(ζ), where ζ is a primitiventh root of unity, is given by

(−1)φ(n)/2nφ(n)∏p|n p

φ(n)/(p−1).

Proposition 5.2.7. Let ζ be a primitive nth root of unity, n > 2 and let Q[ζ]+ =Q[ζ + ζ−1]. Assume n is a prime power, then every unit u ∈ Q[ζ] can be writtenas u = ζ · v with ζ a root of unity and v a unit in Q[ζ]+.

26

Chapter 6

Splitting of Primes in ExtensionFields

6.1 LIFTING OF PRIME IDEALS

For the remainder of time, A is a Dedekind domain with fraction field K, L is afinite, separable extension of K of degree n, and B is the integral closure of A inL.

Definition 6.1.1. Let p be a nonzero prime ideal of A. The lifting (also calledthe extension) of p to B is the ideal pB. Although pB need not be a prime ideal ofB, we can use the fact that B is a Dedekind domain and the unique factorizationtheorem to write

pB =

g∏i=1

Peii

where the Pi are distinct prime ideals of B and the ei are positive integers.

Definition 6.1.2. We can start with a nonzero prime ideal P of B and form aprime ideal of A via

p = P ∩ AWe say that P lies over p, or that p is the contraction of P to A.

Proposition 6.1.3. Let P be a nonzero prime ideal of B. Then P appears in theprime decomposition of pB if and only if P ∩ A = p.

Definition 6.1.4. If we lift p to B and factor pB as∏g

i=1 Peii , the positive integer

ei is called the ramification index of Pi over p (or over A). We say that pramifies in B (or in L) if ei > 1 for at least one i. We have that B/Pi is a finiteextension of the field A/p. The degree fi of this extension is called the relativedegree, residue class degree, inertial degree of Pi over p (or over A).

Proposition 6.1.5. We can identify A/p with a subfield of B/Pi, and B/Pi is afinite extension of A/p.

27

CHAPTER 6. SPLITTING OF PRIMES IN EXTENSION FIELDS

The same argument, with Pi replaced by pB, shows that B/pB is a finitelygeneratedA/p-algebra, in particular, a finite-dimensional vector space overA/p.We will denote the dimension of this vector space by [B/pB : A/p].

Theorem 6.1.6. We have the following identity:g∑i=1

eifi = [B/pB : A/p] = n.

As an example, let d ∈ Z be square-free, L = Q(√d) and let p be a prime

number. The formula above implies g ≤ 2 and we have the following threepossibilities:

(i) g = 2, e1 = e2 = 1, f1 = f2 = 1;in this case we say that p splits in L.(ii) g = 1, e1 = 1, f1 = 2;in this case we say that p remains prime in L.(iii) g = 1, e1 = 2, f1 = 1;this means that p ramifies in L.

Definition 6.1.7. When d is a nonzero square in Fp (resp. is not a square in Fp),we say that d is a quadratic residue (resp. non-residue) modulo p.

Proposition 6.1.8. Let L = Q(√d), the quadratic field associated with the

square-free integer d.(i) The odd primes p for which d is a quadratic residue mod p split in L. So

does 2, if d ≡ 1 mod 8.(ii) The odd primes p for which d is not a quadratic residue mod p remain

prime in L. So does 2, if d ≡ 5 mod 8.(iii) The odd prime divisors of d ramify in L. So does 2, if d ≡ 2 or 3 mod 4.

Proposition 6.1.9. We also have that the ring B/pB ∼=∏g

i=1BPeii .

Theorem 6.1.10. Let L be a number field of degree n over Q, and assume that thering B of algebraic integers of L is Z[θ] for some θ ∈ B. Thus 1, θ, θ2, . . . , θn−1

form an integral basis of B. Let p be a rational prime, and let f be the minimalpolynomial of θ over Q. Reduce the coefficients of f modulo p to obtain f ∈Z[X]. Suppose that the factorization of f into irreducible polynomials over Fpis given by

f = he11 · · ·herr .Let fi be any polynomial in Z[X] whose reduction mod p is hi. Then the ideal

Pi = (p, fi(θ))

is prime, and the prime factorization of (p) in B is

(p) = P e11 · · ·P er

r .

28


6.2 THE DISCRIMINANT AND RAMIFICATION

Definition 6.2.1. We call the principal ideal of A generated by the discriminantof any base of B over A the discriminant of B over A and denote it DB/A.

Lemma 6.2.2. Let A be a ring and let B1, . . . , Bg be rings containing A whichare free A-modules of finite type, and let B =

∏gi=1Bi be the product ring. Then

DB/A =∏g

i=1 DBi/A.

Lemma 6.2.3. Let B be a ring, A a subring of B, and a an ideal of A. Assumethat B is a free-module over A with the base (x1, . . . , xn). For x ∈ B write x forthe residue class of x in B/aB. Then (x1, . . . , xn) is a base of B/aB over A/aand

D(x1, . . . , xn) = D(x1, . . . , xn).

Lemma 6.2.4. Let K be a field which is finite or of characteristic zero. Let L bea finite dimensional (commutative) K-algebra. In order that L be reduced it isnecessary and sufficient that DL/K 6= 0.

Definition 6.2.5. Let K and L be number fields with K ⊂ L. The discriminantof B over A (or of L over K) is the ideal of A generated by the discriminants ofbases of L over K which are contained in B. Denoted DB/A or DL/K .

Theorem 6.2.6. In order that a prime ideal p of A ramify in B, it is necessaryand sufficient that it contain the discriminant DB/A. There are only finitely manyprime ideals of A which ramify in B.

Proposition 6.2.7. If L|K is seperable, then there are only finitely many primeideals of K which are ramified in L.

6.3 QUADRATIC RECIPROCITY

Definition 6.3.1. Given an odd prime p and an integer d relatively prime to p,we define the Legendre symbol as follows:

(dp

)= +1, if d is a quadratic residue mod p,(

dp

)= −1, if d is a non-residue mod p

The multiplicative group F∗p being cyclic of even order p−1, the squares in F∗pform a subgroup (F∗p)2 of index 2 and F∗p/(F∗p)2 is isomorphic to +1,−1. TheLegendre symbol stands for the compositon of the following homomorphisms:

Z→ pZ→ F∗p → F∗p/(F∗p)2 ∼= +1,−1.

29


As a consequence there is the formula:(ab

p

)=

(a

p

)(b

p

), with a, b ∈ Z− pZ.

Proposition 6.3.2. (Euler’s Criterion) If p is an odd prime and if p - a, then(a

p

)≡ a(p−1)/2 mod p.

Theorem 6.3.3. (The Law of Quadratic Reciprocity) If p and q are distinct oddprime numbers, then (

p

q

)(q

p

)= (−1)(p−1)(q−1)/4.

Proposition 6.3.4. (The Complementary Formulas) If p is an odd prime, then(i)(−1p

)= (−1)(p−1)/2 and

(ii)(

2p

)= (−1)(p2−1)/8.

Proposition 6.3.5. (The First Supplementary Law) Since (p − 1)/2 is even ifp ≡ 1 mod 4 and odd if p ≡ 3 mod 4, then we have(

−1

p

)= (−1)(p−1)/2 =

1 if p ≡ 1 mod 4

−1 if p ≡ 3 mod 4.

Proposition 6.3.6. (The Second Supplementary Law)(2

p

)= (−1)(p2−1)/8

so (2

p

)=

1 if p ≡ ±1 mod 8

−1 if p ≡ ±3 mod 8.

Proposition 6.3.7. (Fermat) Any prime number p ≡ 1 mod 4 may be repre-sented as the sum of two squares.

Theorem 6.3.8. (Lagrange) Every natural number may be represented as thesum of four squares. Hence any prime number is the sum of four squares.

Theorem 6.3.9. If g is a primitive root mod p, then g is a quadratic nonresiduemod p. Consequently, exactly half of the integers 1, 2, . . . , p − 1 are quadraticresidues and half quadratic nonresidues.

Definition 6.3.10. Let K be a field of characteristic 6= p such that K contains thepth roots of unity. Let ζ ∈ K be a primitive pth root of unity. Define the Gausssum by

τp =

p−1∑a=1

(a

p

)ζa.

30


Theorem 6.3.11. We have that τ 2p = (−1)(p−1)/2p.

Definition 6.3.12. Let Q be an odd positive integer with prime factorizationQ = q1 · · · qs. The Jacobi symbol is defined by(

a

Q

)=

s∏i=1

(a

qi

).

It follows directly from the definition that(a

Q

)(a

Q′

)=

(a

QQ′

),

(a

Q

)(a′

Q

)=

(aa′

Q

).

Also

a ≡ a′ mod Q⇒(a

Q

)=

(a′

Q

)because if a ≡ a′ mod Q then a ≡ a′ mod qi for i = 1, . . . , s.

Theorem 6.3.13. If Q is an odd positive integer then(−1

Q

)= (−1)(Q−1)/2

Theorem 6.3.14. If Q is an odd positive integer, then(2

Q

)= (−1)(Q2−1)/8

Theorem 6.3.15. If P and Q are odd, relatively prime positive integers, then(P

Q

)(Q

P

)= (−1)(P−1)(Q−1)/4

31


32

Chapter 7

Galois Extensions

7.1 DECOMPOSITION AND INERTIA GROUPS

We still have the standard setup. A is a Dedekind domain with fraction field K,L is a finite separable extension of K, and B is the integral closure of A in L.Now we add the condition that the extension L/K is normal, hence Galois. Wewill let G denote the Galois group Gal(L/K).

Proposition 7.1.1. If σ ∈ G, then σ(B) = B. If P is a prime ideal of B, then sois σ(P). Moreover, if P lies above the nonzero prime ideal p of A, then so doesσ(P). Thus G acts on the set of prime ideals lying above p.

Theorem 7.1.2. Let P and P1 be prime ideals lying above p. Then for someσ ∈ G we have σ(P) = P1.

Corollary 7.1.3. In the factorization pB =∏g

i=1 Peii of the nonzero prime ideal

p, the ramification indices ei are the same for all i, as are the relative degrees fi.Thus we have the simpler identity efg = n, where n = [L : K] = |G|.

In the Galois case, the prime decomposition of p in O takes on the followingsimple form:

p =

(∏σ

σP

)e

where σ varies over a system of representatives of G/GP.

Definition 7.1.4. We say that the prime ideals σ(P), σ ∈ G, are the conjugatesof P.

Definition 7.1.5. The decomposition group of P is the subgroup D of G con-sisting of those σ ∈ G such that σ(P) = P. (This does not mean that σ fixesevery element of P).

33

CHAPTER 7. GALOIS EXTENSIONS

Proposition 7.1.6. Let ZP be the decomposition field, and let PZ = P ∩ ZP bethe prime ideal of ZP below P. Then we have

i) PZ is nonsplit in L, i.e., P is the only prime ideal of L above PZ .ii) P over ZP has ramification index e and inertia degree f .iii) The ramification index and the inertia degree of PZ over K both equal 1.

By the orbit-stabilizer theorem, the size of the orbit of P is the index of thestabilizer subgroup D. Since there is only one orbit, of size g,

g = [G : D] = |G|/|D|, hence |D| = n/g = efg/g = ef,

independent of P. Note also that distinct conjugates of P determine distinctcosets of D. For if σ1D = σ2D, then σ−1

2 σ1 ∈ D, so σ1(P) = σ2(P).There is a particular subgroup of D that will be of interest. We know that

if σ ∈ D, then σ(P) = P. It follows that σ induces an automorphism σ ofB/P. (Note that x ≡ y mod P if and only if σx ≡ σy mod P). Since σ is aK-automorphism, σ is an A/p-automorphism. The mapping σ → σ is a grouphomomorphism from D to the group of A/p-automorphisms of B/P.

Definition 7.1.7. The kernel I of the above homomorphism, that is, the set of allσ ∈ D such that σ is trivial, is called the inertia group of P. So we have

I = σ ∈ D : σ(x)+P = x+P,∀x ∈ B = σ ∈ D : σ(x)−x ∈ P,∀x ∈ B

L B

KD AD

K A

Take KD to be the fixed field of D, and let AD = B ∩KD be the integral closureof A in KD. Let pD be the prime ideal P ∩ AD. Note that P is the only primefactor of pDB. This is because all primes in the factorization are conjugate, andσ(P) = P for all σ ∈ D.

Lemma 7.1.8. Let pDB = Pe′ and f ′ = [B/P : AD/pD]. Then e′ = e andf ′ = f . Moreover, A/p ∼= AD/pD.

Theorem 7.1.9. The homomorphism σ → σ of D to Gal[(B/P)/(A/p)] intro-duced earlier, is surjective with kernel I. Therefore Gal[(B/P)/(A/p)] ∼= D/I .

Corollary 7.1.10. The order of I is e. Thus the prime ideal p does not ramify ifand only if the inertia group of every prime ideal P lying over p is trivial.

We’ll write κ(P) = O/P and κ(p) = o/p.

34


Proposition 7.1.11. Let TP be the fixed field of the inertia group, and ZP thefixed field of the decomposition group. The extension TP|ZP is normal, and onehas

G(TP|ZP) ∼= G(κ(P)|κ(p)), G(L|TP) = IP.

If the residue field extension κ(P)|κ(p) is separable, then one has

|IP| = [L : TP] = e, (GK : IP) = [TP : ZP] = f

In this case one finds for the prime ideal PT of TP below P:i) The ramification index of P over PT is e and the inertia degree is 1.ii) The ramification index of PT over PZ is 1, and the inertia degree is f .

7.2 THE FROBENIUS AUTOMORPHISM

We now assume that K and L are number fields.

Definition 7.2.1. Let p be a prime ideal of A that does not ramify in B, and letP be a prime lying over p. We know I(P) is trivial, so Gal[(B/P)/(A/p)] ∼=D(P). But B/P is a finite extension field of A/p, so the Galois group is cyclic.There is a generator given by x + P → xq + P, x ∈ B, where q = |A/p|. Wehave identified an element σ ∈ D(P), called the Frobenius automorphism orsimply the Frobenius of P relative to L/K. The Frobenius automorphism isdetermined by the requirement that for every x ∈ B,

σ(x) ≡ xq mod P

We use the notation[L/KP

]for the Frobenius automorphism.

Proposition 7.2.2. If τ ∈ G, then[L/Kτ(P)

]= τ

[L/KP

]τ−1.

Corollary 7.2.3. If L/K is abelian, then[L/KP

]depends only on p, and we write

the Frobenius automorphism as(L/Kp

), and sometimes call it the Artin symbol.

We now introduce an intermediate field between K and L, call it F. We canthen lift p to the ring of algebraic integers in F, namely B ∩ F . A prime ideallying over p has the form P ∩ F , where P is a prime ideal of pB. We willcompare decomposition groups with respect to the fields L and F, with the aid ofthe identity

[B/P : A/p] = [B/P : (B ∩ F )/(P ∩ F )][(B ∩ F )/(P ∩ F ) : A/p]

The term on the left is the order of the decomposition group of P over p, denotedby D(P, p). (We are assuming that p does not ramify, so e = 1.) The first term

35


on the right is the order of the decomposition group of P over P∩F . The secondterm on the right is the relative degree of P ∩ F over p, call it f. Thus

|D(P,P ∩ F )| = |D(P, p)|/f

Since D = D(P, p) is cyclic and is generated by the Frobenius automorphismσ, the unique subgroup of D with order |D|/f is generated by σf . Note thatD(P,P ∩ F ) is a subgroup of D(P, p), because Gal(L/F ) is a subgroup ofGal(L/K). It is natural to expect that the Frobenius autormorphism of P, rela-tive to L/F , is σf .

Proposition 7.2.4. We have that[L/FP

]=[L/KP

]f.

Proposition 7.2.5. If the extension F/K is Galois, then the restriction of σ =[L/KP

]to F is

[F/KP∩F

].

Definition 7.2.6. We may view the lifting from the base field K to the extensionfield L as occuring in three distinct steps. Let FD be the decomposition field ofthe extension, that is, the fixed field of the decomposition group D, and let FI bethe inertia field, the fixed field of the inertia group I.

L

FI

FD

K

e = |I|

f = |D|/e

g = n/ef

All ramification takes place at the top, and all splitting at the bottom. There isinertia in the middle. As we move up the diagram, we multiply the ramificationindices and relative degrees. The basic point is that if P = Pe1

1 · · · and P1 =Pe2

2 · · · , then P = Pe1e22 · · · . The multiplicativity of f follows because f is a

vector space dimension.

Proposition 7.2.7. Let q be an odd prime, and let L = Q(ζq) be the cyclotomicfield generated by a primitive qth root of unity. Then L has a unique quadraticsubfield F. Explicitly, if q ≡ 1 mod 4, then the quadratic subfield is Q(

√q), and

if q ≡ 3 mod 4, it is Q(√−q).

36

Chapter 8

The Theory of Valuations

8.1 THE P-ADIC NUMBERS

First we define the local ring, for a prime p

Z(p) =gh

: g, h ∈ Z, p - h

Definition 8.1.1. Fix a prime number p. A p-adic integer is a formal infiniteseries

a0 + a1p+ a2p2 + · · · ,

where 0 ≤ ai < p, for i = 0, 1, . . . The set of all p-adic integers is denoted Zp.

Proposition 8.1.2. The residue classes a mod pn ∈ Z/pnZ can be uniquelyrepresented in the form

a ≡ a0 + a1p+ a2p2 + · · ·+ an−1p

n−1 mod pn

where 0 ≤ ai < p for i = 0, . . . , n− 1.

Definition 8.1.3. Every integer f and, more generally, every rational numberf ∈ Z(p) defines a sequence of residue classes

sn = f mod pn ∈ Z/pnZ, n = 1, 2, . . .

The sequence of numbers

sn = a0 + a1p+ a2p2 + · · ·+ an−1p

n−1, n = 1, 2, . . .

defines a p-adic integer∞∑v=0

avpv ∈ Zp.

We call it the p-adic expansion of f.

37

CHAPTER 8. THE THEORY OF VALUATIONS

Definition 8.1.4. In analogy with the Laurent series f(z) =∑∞

v=−m av(z− a)v,we now extend the domain of p-adic integers into that of the formal series

∞∑v=−m

avpv = a−mp

−m + · · ·+ a−1p−1 + a0 + a1p+ · · · ,

where m ∈ Z and 0 ≤ av < p. Such series we call p-adic numbers and wewrite Qp for them.

If f ∈ Q then we write

f =g

hp−m where g, h ∈ Z, (gh, p) = 1,

and ifa0 + a1p+ a2p

2 + · · ·is the p-adic expansion of g

h, then we attach to f the p-adic number

a0p−m + a1p

−m+1 + · · ·+ am + am+1p+ · · · ∈ Qp

as its p-adic expansion. In this way we obtain a canonical mapping Q → Qp,which takes Z into Zp and is injective. For if a, b ∈ Z have the same p-adicexpansion, then a − b is divisible by pn for every n and hence a = b. Onecan define addition and multiplication of p-adic numbers which turn Zp into aring, and Qp into its field of fractions. We will represent the p-adic numbers assequences of residue classes

sn = sn mod pn ∈ Z/pnZ.The terms of such a sequence lie in different rings, but these are related by thecanonical projections

Z/pZ λ1←− Z/p2Z λ2←− Z/p3Z λ3←− · · ·and we find that λn(sn+1) = sn. In the direct product

∞∏n=1

Z/pnZ = (xn)n∈N : xn ∈ Z/pnZ.

we now consider all elements (xn) with the property that

λn(xn+1) = xn for all n = 1, 2, . . .

This set is called the projective limit of the rings Z/pnZ and is denoted lim← Z/pnZ.In other words, we have

lim←

Z/pnZ = (xn) ∈∞∏n=1

Z/pnZ : λn(xn+1) = xn, n = 1, 2, . . .

Definition 8.1.5. We identify Zp with lim← Z/pnZ and obtain the ring of p-adicintegers Zp.

38


8.2 THE P-ADIC ABSOLUTE VALUE

Definition 8.2.1. The p-adic absolute value | · |p is defined as follows. Leta = b

c, b, c ∈ Z be a nonzero rational number. We extract from b and from c as

high a power of the prime number p as possible

a = pmb′

c′, (b′c′, p) = 1

and we put

|a|p =1

pm.

The exponent m in the representation of the number a is denoted by vp(a),and one puts formally vp(0) =∞. This gives the function

vp : Q→ Z ∪ ∞,

which satisfies the properties(i) vp(a) =∞⇔ a = 0,(ii) vp(ab) = vp(a) + vp(b),(iii) vp(a+ b) ≥ minvp(a), vp(b),

Definition 8.2.2. The function vp is called the p-adic exponential valuation ofQ. The p-adic absolute value is given by

| · |p : Q→ R, a 7→ |a|p = pvp(a).

Proposition 8.2.3. For every rational number a 6= 0, one has∏p

|a|p = 1,

where p varies over all prime numbers as well as the symbol∞.

Proposition 8.2.4. The field Qp of p-adic numbers is complete with respect to| · |p, i.e., every Cauchy sequence in Qp converges with respect to | · |p.

Proposition 8.2.5. The set

Zp = x ∈ Qp : |x|p ≤ 1

is a subring of Qp. It is the closure with respect to | · |p of the ring Z in the fieldQp.

The group of units of Zp is Z∗p = x ∈ Zp : |x|p = 1. Every element x ∈ Q∗padmits a unique representation x = pmu with m ∈ Z and u ∈ Z∗p.

39


Proposition 8.2.6. The nonzero ideals of the ring Zp are the principal ideals

pnZp = x ∈ Qp : vp(x) ≥ n,

with n ≥ 0, and one hasZp/pnZp ∼= Z/pnZ.

Proposition 8.2.7. The homomorphism

Zp → lim←

Z/pnZ

is an isomorphism.

Proposition 8.2.8. There is a canonical isomorphism

Zp ∼= Z [[X]] /(X − p).

Definition 8.2.9. The topology defined by the p-adic absolute valute |·|p is calledthe p-adic topology on K.

8.3 VALUATIONS

Definition 8.3.1. A valuation of a field K is a function

| · | : K → R

with the properties(i) |x| ≥ 0, and |x| = 0⇔ x = 0,(ii) |xy| = |x||y|,(iii) |x+ y| ≤ |x|+ |y|

Definition 8.3.2. Two valuations of K are called equivalent if they define thesame topology on K.

Proposition 8.3.3. Two valuations | · |1 and | · |2 on K are equivalent if and onlyif there exists a real number s > 0 such that one has

|x|1 = |x|s2

for all x ∈ K.

Theorem 8.3.4. (Approximation Theorem) Let | · |1, . . . , | · |n be pairwise in-equivalent valuations of the field K and let a1, . . . , an ∈ K be given elements.Then for every ε > 0 there exists an x ∈ K such that

|x− ai|i < ε for all i = 1, . . . , n.

40


Definition 8.3.5. The valuation |·| is called nonarchimedean if |n| stays boundedfor all n ∈ N. Otherwise it is called archimedean.

Proposition 8.3.6. The valuation | · | is nonarchimedean if and only if it satisfiesthe strong triangle inequality

|x+ y| ≤ max|x|, |y|.

Proposition 8.3.7. Every valuation of Q is equivalent to one of the valuations| · |p or | · |∞.

Theorem 8.3.8. (Ostrowski) Let | · | be a nontrivial absolute value on Q.i) If | · | is archimedean, then | · | is equivalent to | · |∞.ii) If | · | is nonarchimedean, then it is equivalent to | · |p for exactly one prime

p.

Definition 8.3.9. Let K be an algebraic number field. An equivalence class ofabsolute values on K is called a prime or place of K.

Theorem 8.3.10. Let K be an algebraic number field. There exists exactly oneprime of K

i) for each prime ideal p;ii) for each real embedding;iii) for each conjugate pair of complex embeddings.

Definition 8.3.11. We generally write v for a prime. If it corresponds to a primeideal p of OK , then we call it a finite prime, and we write pv for the ideal. Ifit corresponds to a real or nonreal embedding of K, then we call it an infinite(real or complex) prime. We write | · |v for an absolute value in the equivalenceclass. If L ⊃ K and w and v are primes of L and K such that | · |w restrictedto K is equivalent to | · |v, then we say that w divides v, or w lies over v, andwe write w | v. For a finite prime, this means Pw ∩ OK = pv or, equivalently,that Pw divides pv · OL. For an infinite prime, it means that w corresponds toan embedding σ : L → C that extends the embedding corresponding to v (or itscomplex conjugate).

Theorem 8.3.12. (Product Formula) For each prime v, let |·|v be the normalizedabsolute value. For every nonzero α ∈ K,

Π|α|v = 1 (product over all primes of K).

Lemma 8.3.13. Let L be a finite extension of a number field K.i) Each prime on K extends to a finite number of primes of L.ii) For every prime v of K and α ∈ L×,

Πw|v|α|w = |NL/Kα|v.

Definition 8.3.14. Let | · | be a nonarchimedean valuation of the field K. Puttingv(x) = − log |x| for x 6= 0, and v(0) = ∞, we obtain a function v : K →

41


R ∪ ∞. Such a function v on K with the following properties is called anexponential valuation of K.

(i) v(x) =∞⇔ x = 0,(ii) v(xy) = v(x) + v(y),(iii) v(x+ y) ≥ minv(x), v(y).

Two exponential valuations v1 and v2 of K are called equivalent if v1 = sv2

for some real number s > 0. For every exponential valuation v we obtain avaluation by putting

|x| = q−v(x),

for some fiexed real number q > 1. To distinguish it from v, we call | · | anassociated absolute value.

Proposition 8.3.15. The subset

O = x ∈ K : v(x) ≥ 0 = x ∈ K : |x| ≤ 1

is a ring with group of units

O∗ = x ∈ K : v(x) = 0 = x ∈ K : |x| = 1

and the unique maximal ideal

p = x ∈ K : v(x) > 0 = x ∈ K : |x| < 1.

Definition 8.3.16. O is an integral domain with field of fractions K and has theproperty that for every x ∈ K, either x ∈ O of x−1 ∈ O. Such a ring is calleda valuation ring. Its only maximal ideal is p = x ∈ O : x−1 6∈ O. The fieldO/p is called the residue class field of O.

Definition 8.3.17. An exponential valuation v is called discrete if it admits asmallest positive value s. It is normalized if s = 1. An element π ∈ O suchthat v(π) = 1 is a prime element, and every element x ∈ K∗ admits a uniquerepresentation x = uπm.

Proposition 8.3.18. If v is a discrete exponential valuation of K, then

O = x ∈ K : v(x) ≥ 0

is a principal ideal domain, hence a discrete valuation ring. Suppose v is nor-malized. Then the nonzero ideals of O are given by

pn = πnO = x ∈ K : v(x) ≥ n, n ≥ 0,

where π is a prime element, i.e., v(π) = 1. One has

pn/pn+1 ∼= O/p.

42


If v is a normalized exponential valuation and | · | = q−v(q > 1) an associatedmultiplicative valuation, then

pn = x ∈ K : |x| < 1

qn−1

We obtain the descending chain of subgroups

O∗ = U (0) ⊃ U (1) ⊃ · · ·

where

U (n) = 1 + pn = x ∈ K∗ : |1− x| < 1

qn−1, n > 0

Definition 8.3.19. U (n) is called the nth higher unit group and U (1) the groupof principal units.

Proposition 8.3.20. O∗/U (n) ∼= (O/pn)∗ and U (n)/U (n+1) ∼= O/p, for n ≥ 1.

8.4 COMPLETIONS

Definition 8.4.1. A valued field (K, | · |) is called complete if every Cauchysequence ann∈N in K converges to an element a ∈ K, i.e.,

limn→∞

|an − a| = 0.

As usual, we call an a Cauchy sequence if for every ε > 0 there exists N ∈ Nsuch that

|an − am| < ε for all n,m ≥ N.

From any valued field (K, | · |) we get a complete valued field (K, | · |) by theprocess of completion. We call an a nullsequence if limn→∞ an = 0. Takethe ring R of all Cauchy sequences of (K, | · |), consider therein the maximalideal m of all nullsequences with respect to | · |, and define

K = R/m.

One embeds the field K into K by sending every a ∈ K to the class of theconstant Cauchy sequence (a, a, . . .). The valuation | · | is extended from K toK by giving the element a ∈ K which is represented by the Cauchy sequencean the absolute value

|a| = limn→∞

|an|.

43


Proposition 8.4.2. If o ⊂ K, resp o ⊂ K, is the valuation ring of v, resp. of v,and p, resp p, is the maximal ideal, then one has

o/p ∼= o/p

and if v is discrete, one has furthermore

o/pn ∼= o/pn for n ≥ 1.

Generalizing the p-adic expansion to the case of an arbitrary discrete valua-tion v of the field K, we have the following:

Proposition 8.4.3. Let R ⊂ o be a system of representatives for κ = o/p suchthat 0 ∈ R, and let π ∈ o be a prime element. Then every x 6= 0 in K admits aunique representation as a convergent series

x = πm(a0 + a1π + a2π2 + · · · )

where ai ∈ R, a0 6= 0,m ∈ Z.

Definition 8.4.4. Let K be a field which is complete with respect to a nonar-chimedean valuation | · |. Let o be the corresponding valuation ring with max-imal ideal p and residue class field κ = o/p. We call a polynomial f(x) =a0 + a1x+ · · ·+ anx

n ∈ o[x] primitive if f(x) 6≡ 0 mod p, i.e., if

|f | = max|a0|, . . . , |an| = 1.

Lemma 8.4.5. (Hensel’s Lemma) If a primitive polynomial f(x) ∈ o[x] admitsmodulo p a factorization

f(x) ≡ g(x)h(x) mod p

into relatively prime polynomials g, h ∈ κ[x], then f(x) admits a factorization

f(x) = g(x)h(x)

into polynomials g, h ∈ o[x] such that deg(g) = deg(g) and

g(x) ≡ g(x) mod p and h(x) ≡ h(x) mod p.

Definition 8.4.6. A global field is a finite extension of either Q or Fp(t). Com-pletions of global fields have the most eminent relevance for number theory. Allfields which are complete with respect to a discrete valuation and have a finiteresidue class field are called local fields. They are the finite extensions of Qp andFp ((t)) (power series field). The finite extensions K|Qp of the fields of p-adicnumbers Qp are called p-adic number fields.

Theorem 8.4.7. Let K be complete with respect to a discrete absolute value| · |K , and let L be a finte separable extension of K of degree n. Then | · |Kextends uniquely to a discrete absolute value | · |L on L, and L is complete forthe extended absolute value. For all β ∈ L,

|β|L = |NL/Kβ|1/nK

44


Corollary 8.4.8. LetK be as previous theorem, and let Ω be a (possibly infinite)separable algebraic extension of K. Then | · |K extends in a unique way to anabsolute value | · |Ω on Ω.

Proposition 8.4.9. LetK be complete with respect to a nonarchimedean discreteabsolute value. LetA be the ring of integers inK and letm be the maximal idealin A. Then A is compact if and only if A/m is finite.

Corollary 8.4.10. Assume that the residue field is finite. Then pn, 1 + pn, andA× are all compact.

Definition 8.4.11. A local field is a fieldK with a nontrivial absolute value suchthat K is locally compact (and hence complete).

8.5 UNRAMIFIED AND TAMELY RAMIFIED EXTENSIONS

Definition 8.5.1. A henselian field is a field with a nonarchimedean valuation vwhose valuation ring o satisfies Hensel’s lemma. One also calls the valuation vor the valuation ring o henselian.

In this section we fix a base field K which is henselian with respect to anonarchimedean valuation v. As before, we denote the valuation ring, the max-imal ideal and the residue class field by o, p, κ, respectively. If L|K is an al-gebraic extension, then the corresponding invariants are labelled w,O,P, λ, re-spectively. Or also, we assume that K is complete under a discrete valuation,with ring A and maximal ideal p.

Definition 8.5.2. An extension L of a number field K is said to be unramifiedover K if no prime ideal of OK ramifies in OL.

Theorem 8.5.3. There does not exist an unramified extension of Q.

There may exist unramified extensions of number fields other than Q. From classfield theory, the maximal abelian unramified extension of K, called the Hilbertclass field of K, has Galois group isomorphic to Cl(OK).

Definition 8.5.4. A finite extension L|K is called unramified if the extensionλ|κ of the residue class field is seperable and one has

[L : K] = [λ : κ].

An arbitrary algebraic extension L|K is called unramified if it is a union of finiteunramified subextensions.

45


Definition 8.5.5. We see that e = 1 if and only if

[E : K] = [B/P : A/p].

If this equality holds and the residue class field extension B/P over A/p isseparable, then we say that P is unramified over p, or that E is unramified overK.

Let φ : B → B/P be the canonical homomorphism. If g = βnXn + · · ·+β0

is a polynomial with coefficients in B, then we denote by gφ the polynomialφ(βn)Xn + · · ·+ φ(β0), obtained by φ.

Proposition 8.5.6. Let E be finite overK, and assume that P is unramified overp. Let a ∈ Bφ be such that Bφ = Aφ(a) and let a be an element of B such thatφa = a. Then E = K(a), and the irreducible poly g(X) of a over K is suchthat gφ is irreducible. Conversely, if E = K(a) for some a ∈ B satisfying a polyg(X) inA(X) having leading coefficient 1 and such that gφ has no multiple root,then P is unramified over p and Bφ = Aφ(φa).

Proposition 8.5.7. Let E be a finite extension of K.i) If E ⊃ F ⊃ K, then E is unramified over K if and only if E is unramified

over F and F is unramified over K.ii) If E is unramified over K, and K1 is a finite extension of K, then EK1 is

unramified over K1.iii) If E1 and E2 are finite unramified over K, then so is E1E2.

Corollary 8.5.8. LetK be a p-adic field (i.e., completion of a number field undera p-adic valuation). Let E be an unramified extension of K. Then every unit ofK is a norm of a unit in E.

We still assume thatK is complete, under a discrete valuation, with Dedekindring A and maximal ideal p, and we assume that A/p is perfect. If E is a finiteextension, we denote by B = BE the integral closure of A in E, and P = PE

its maximal ideal.

Definition 8.5.9. We say that P is totally ramified above p if [E : K] = e. Inthat case, the residue class degree is equal to 1 (because ef = n). Since P isthe only prime of B lying above p, we say that E is totally ramified over K.

Proposition 8.5.10. Let E be a finite extension of K. Let Eu be the composite ofall unramified subfields over K. Then Eu is unramified over K, and E is totallyramified over Eu.

Definition 8.5.11. Let E be a finite extension of K. We say that P is tamelyramified over p (or E tamely ramified over K) if the characteristic p of theresidue class field A/p does not divide e. If it does, we say that P is stronglyramified.

46


Proposition 8.5.12. Assume that E is totally ramified over K. Let Π be an ele-ment of order 1 at P. Then Π satisfies an Eisenstein equation Xe + ae−1X

e−1 +· · · + a0 = 0, where ai ∈ p for all i and a0 6≡ 0 mod p2. Conversely, suchan equation is irreducible, and a root generates a totally ramified extension ofdegree e.

Proposition 8.5.13. Let E be totally and tamely ramified over K. Then thereexists an element Π of order 1 at P in E satisfying an equation Xe−π = 0 withπ of order 1 at p in K. Conversely, let a be an element of A, and e a positiveinteger not divisible by p. Then any root of an equation Xe − a = 0 generates atamely ramified extension of K, and this extension is totally ramified if the orderat p of a is relatively prime to e.

Proposition 8.5.14. Let E be a finite extension of K.i) If E ⊃ F ⊃ K, then E is tamely ramified over K if and only E is tamely

ramified over F and F is tamely ramified over K.ii) If E is tamely ramified over K, and K1 is a finite extension of K, then

EK1 is tamely ramified over K1.iii) If E1 and E2 are finite tamely ramified over K, then so is E1E2.

Corollary 8.5.15. LetE be a finite extension ofK, and letEt be the compositumof all tamely ramified subextensions. Then Et is tamely ramified over K, and Eis totally ramified over Et. Furthermore, if p is the characteristic of the residueclass field, then the degree [E : Et] is a power of p.

Proposition 8.5.16. Let K be a p-adic field (finite extension of Qp). Given aninteger n, there exists only a finite number of extensions of degree ≤ n.

Let L be a finite Galois extension of K, and assume that the residue field k ofK is perfect. We have that Gal(L/K) preserves the absolute value on L. Inparticular, it preserves

B = α ∈ L : |α| ≤ 1, p = α ∈ L : |α| < 1.

Let π be a prime element of L (so that p = (π)). We define a sequence ofsubgroups G ⊃ G0 ⊃ G1 ⊃ · · · by the condition

σ ∈ Gi ⇔ |σα− α| < |π|i all α ∈ B.

Definition 8.5.17. The group G0 is called the inertia group, the group G1 iscalled the ramification group, and the group Gi, i > 1, are called the higherramification groups of L over K.

Theorem 8.5.18. Let L/K be a Galois extension, and assume that the residuefield extension l/k is separable.

47


i) The fixed field of G0 is the largest unramified extension K0 of K in L, and

G/G0 = Gal(K0/K) = Gal(l/k).

ii) For i ≥ 1, the group

Gi = σ ∈ G0 : |σπ − π| < |π|i.

Corollary 8.5.19. We have an exhaustive filtration G ⊃ G0 ⊃ · · · such thatG/G0 = Gal(l/k);G0/G1 → l×;Gi/Gi+1 → l.Therefore, if k is finite, then Gal(L/K) is solvable.

Definition 8.5.20. An extension L/K is said to be wildly ramified if p | e wherep = char(k). Otherwise it is said to be tamely ramified.

Hence, for a Galois extension

L/K is unramified ⇔ G0 = 1,

andL/K is tamely ramified ⇔ G1 = 1.

Proposition 8.5.21. Let L = K(ζ), and let O|o, resp, λ|κ, be the extension ofvaluation rings, resp, residue class fields of L|K. Suppose that (n, p) = 1. Thenone has:

i) The extension L|K is unramified of degree f , where f is the smallest natu-ral number such that qf ≡ 1 mod n.

ii) The Galois group G(L|K) is canonically isomorphic to G(λ|κ) and isgenerated by the automorphism φ : ζ 7→ ζq.

iii) O = o[ζ].

Proposition 8.5.22. Let ζ be a primitive pm-th root of unity. Then one has:i) Qp(ζ)|Qp is totally ramified of degree φ(pm) = (p− 1)pm−1.ii) G(Qp(ζ)|Qp) ∼= (Z/pmZ)∗.iii) Zp[ζ] is the valuation ring of Qp(ζ).iv) 1− ζ is a prime element of Zp[ζ] with norm p.

48


8.6 GLOBAL FIELDS

Proposition 8.6.1. Let L = K[α] be a finite separable extension of K, and letf(X) be the minimum polynomial of α over K. Then there is a natural one-to-one correspondence between the extensions of | · | to L and the irreduciblefactors of f(X) in K[X].

Proposition 8.6.2. Let | · | be a absolute value on K (archimedean or discretenonarchimedean) and let L be a finite separable extension of K. Let K be thecompletion of K with respect to | · |. Then | · | has finitely many extensions| · |1, . . . , | · |g to L; if Li denotes the completion of L with respect to the absolutevalue | · |i, then L⊗K K ∼= Πg

i=1Li.

Corollary 8.6.3. In the situation of the proposition, for any α ∈ L,

NL/K(α) = ΠNLi/K(α), T rL/K(α) =

∑TrLi/K

(α).

where the ith factor or summand on the right, α is regarded as an element of Li.

Proposition 8.6.4. LetL/K be a finite extension of number fields. For any primev of K and α ∈ L,

Πw|v‖α‖w = ‖NL/Kα‖v.

Here ‖ · ‖ denotes the normalized absolute values for the primes w and v.

Let L be a finite Galois extension of a number field K. For a absolute value wof L, we write σw for the absolute value such that |σα|σw = |α|w, i.e., |α|σw =|σ−1α|w. For example, if w is the prime defined by a prime ideal P, then σw isthe prime defined by the prime ideal σP, because

|α|σw < 1⇔ σ−1α ∈ P⇔ α ∈ σP.

Definition 8.6.5. The group G acts on the set of primes of L lying over a fixedprime v of K, and we define the decomposition (splitting) group of w to be thestabilizer of w in G; thus

Gw = σ ∈ G : σw = w.

Equivalently,Gw is the set of elements ofG that act continuously for the topologydefined by | · |w. Each σ ∈ Gw extends uniquely to a continuous automorphismof Lw.

Proposition 8.6.6. The homomorphism Gw → Gal(Lw/Kv) is an isomorphism.

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LetD(P) orG(P) be the decomposition group of P, so thatD(P) = Gal(LP/Kp).Let I(P) ⊂ D(P) be the inertia group.

Proposition 8.6.7. For PI = P ∩ LI(P),PD = P ∩ LD(P), p = P ∩K;i) The only prime ideal of L lying over PD is P.ii) The prime ideal PD is unramified in LI , and f(PI/PD) = f(P/p).iii) The prime ideal PI is totally ramified in L, and e(P/PI) = e(P/p).iv) If D(P) is normal in G, then

pOLD = ΠσPD

where the product is over a set of representatives for G/D(P).

Proposition 8.6.8. AssumeM is Galois overK with Galois groupG, and thatHis the subgroup ofG fixingL. Let P be a prime ideal inOM , and let PL = P∩L.

i) The decomposition group H(P) of P over L is G(P) ∩H .ii) Suppose further that H is a normal subgroup of G, so that G/H is the

Galois group of L/K. The decomposition group of PL over K is the image ofG(P) in G/H .

50

Chapter 9

Analytical Methods

9.1 THE CLASSICAL ZETA-FUNCTIONS

Definition 9.1.1. The Dedekind zeta-function of an algebraic number field Kis defined by

ζK(s) =∑I

1

N(I)s

running over the nonzero ideals of K.

Proposition 9.1.2. In the half-plane σ > 1, the previous series converges abso-lutely, and the convergence is uniform in every compact subset of that half-plane,so ζK(s) is regular there. Moreover, in that half-plane the infinite product

P (s) =∏p

(1− 1

N(p)s

)−1

converges, and we have P (s) = ζK(s).

Corollary 9.1.3. The function ζK(s) does not vanish in the half plane σ > 1.

Corollary 9.1.4. The function ζK(s)−1 is regular for σ > 1, and we have therethe equality

1

ζK(s)=∑I

µK(I)

N(I)s

Moreover, in the same range on has

|ζK(s)−1| ≤ ζK(σ).

Corollary 9.1.5. If F (n) denotes the number of ideals of RK with norm equalto n, then for σ > 1 one has

ζK(s) =∞∑n=1

F (n)

ns

51

CHAPTER 9. ANALYTICAL METHODS

and|ζK(s)| ≤ ζ(σ)N

where N = [K : Q], and ζ(s) = ζQ(s) is the Riemann zeta function.

Definition 9.1.6. WithR(K) being the regulator of K, d(K) the disciminant andw(K) the number of roots of unity lying in K, we define

K =2r1(2π)r2R(K)

w(K)√|d(K)|

Theorem 9.1.7. The function ζK(s) can be continued analytically to a mero-morphic function having a unique simple pole at s = 1 with the residue hK.Moreover, if we put N = [K : Q], and

A =1

2r2πN/2

√|d(K)|,

then the functionΦ(s) = A2Γ(s/2)r1Γ(s)r2ζK(s)

satisfies the functinal equation Φ(s) = Φ(1− s).

Corollary 9.1.8. For every field K of degree n we have

h(K)R(K) = O(√D logn−1D),

with D = |d(K)|, the implied constant depending only on n.

Corollary 9.1.9. If K and L are number fields with the same Dedekind zeta-function, then [K : Q] = [L : Q], d(K) = d(L), r1(K) = r1(L), r2(K) =r2(L), and

h(K)R(K)

w(K)=h(L)R(L)

w(L).

Corollary 9.1.10. If K is either Q, or an imaginary quadratic field, then ζK(0) 6=0. Otherwise ζK(s) has a zero of order r1 + r2 − 1 at s = 0.

Definition 9.1.11. For an arbitrary Hecke character χwe define the Hecke zeta-function by the formula

ζ(s, χ) =∑I

χ(I)

N(I)s,

the sum taken over all nonzero ideals of RK .

Proposition 9.1.12. If χ(I) = χ1(I)N(I)w with a normalized Hecke characterχ1 and a complex w, then the series defining ζ(s, χ) converges absolutely in the

52


half plane σ > 1 + <w, and defines there a regular function. Moreover in thathalf-plane we have

ζ(s, χ) =∏p

(1

1− χ(p)N(p)−s

),

andζ(s, χ) = ζ(s− w, χ1).

Corollary 9.1.13. Let f be a primitive Dirichlet character mod N and let χ bethe corresponding Hecke character. Assume N > 1 and write N =

∏ri=1 p

aii

with all ai’s positive. Then the function defined for σ > 1 by

L(s, χ) =∞∑n=1

χ(n)

ns,

extends to an integral function, satisying the functional equation.

Corollary 9.1.14. Dedekind’s zeta-function ζK and Dirichlet’s L-functions donot vanish on the line σ = 1.

9.2 ASYMPTOTIC DISTRIBUTION OF IDEALS AND PRIME

IDEALS

Theorem 9.2.1. (Dirichlet’s Prime Number Theorem) In every arithmetic pro-gression

a, a±m, a± 2m, a± 3m, . . . ,

a,m ∈ N, (a,m) = 1, there occur infinitely many prime numbers.

Definition 9.2.2. Let M be a set of prime ideals of K. The limit

d(M) = lims→1+0

∑p∈M N(p)−s∑p N(p)−s

provided it exists, is called the Dirichlet density of M.

Definition 9.2.3. Let S be a set of finite primes in a number field K, and let Pbe the set of all finite primes. We say that S has natural density δ if

limN→∞

|p ∈ S : Np ≤ N||p : Np ≤ N|

= δ.

Theorem 9.2.4. (Chebotarev Density Theorem) Let L be a finite Galois exten-sion of the number field K, with Galois group G, and let C be a conjugacyclass in G. The set of prime ideals p of K such that (p, L/K) = C has densityδ = |C|/|G|.

53


Corollary 9.2.5. The primes that split in L have density 1/[L : K]. In particular,there exist infinitely many primes of K not splitting in L.

Definition 9.2.6. If A is a set of prime ideals and for σ > 1 we have∑p∈A

1

N(p)s= a log

1

s− 1+ g(s)

where g(s) is a regular function in the closed half plane σ > 1, then we say thatA is a regular set of prime ideals and call a its Dirichlet density.

Corollary 9.2.7. The set of all prime ideals of RK is regular, and its Dirichletdensity equals 1.

Corollary 9.2.8. If L/K is a finite extension, and A is the set of all prime idealsof RL of degree 1 over K, then A is regular, and its Dirichlet density equals 1.

Corollary 9.2.9. Let L/K be a normal extension of degree N, and let A be theset of all prime ideals of RK which split in L/K, i.e., which become products ofN distinct prime ideals of the first degree in L. Then A is regular and its Dirichletdensity is 1/N .

Corollary 9.2.10. If L/K is of degree N ≥ 2, then there exist infinitely manyprime ideals of RK which do not split in L/K. If, moreover, L/K is normal,then such ideals form a regular set of Dirichlet density 1− 1/N .

Theorem 9.2.11. (The Prime Ideal Theorem) If πK(x) denotes the number ofprime ideals of RK with norms not exceeding x, then

πK(x) = (1 + o(1))x

log x.

Corollary 9.2.12. If L/K is a normal extension of degree N, and AL/K(x) de-notes the number of prime ideals of RK splitting in L/K, and having norms notexceeding x, then

AL/K(x) =

(1

N+ o(1)

)x

log x.

Theorem 9.2.13. (Ideal Theorem) If M(x) is the number of ideals of RK withnorms bounded by x, then

M(x) = (hK + o(1))x

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