Algebra 3: Exercises

Hans Sterk

Fall 2003

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Contents

1 Exercises 11.1 Week 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Week 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Week 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.4 Week 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.5 Week 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261.6 Week 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.7 Week 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

1.7.1 Resultants . . . . . . . . . . . . . . . . . . . . . . . . . 331.7.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . 35

1.8 Week 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381.9 Week 9 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

1

2 CONTENTS

Chapter 1

Exercises

1.1 Week 1

In the exercises below, Fq denotes a finite field with q elements. Some basicfacts concerning finite fields:

• q is a prime power, i.e. of the form pa for some prime p and positiveinteger a.

• Every prime power occurs as the cardinality of some finite field; an‘explicit’ construction of a field of order pn is as follows: F[X]/(f),where f ∈ F[X] is an irreducible polynomial of degree n.

• Finite fields with the same number of elements are isomorphic (‘up toisomorphism there is only one field of a given prime power’).

• F∗q := Fq\{0} is a cyclic group (with respect to multiplication) of order

q − 1.

• If q = pa, then Fq contains a subfield of order q′ if and only if q′ = pb

for some b that divides a. In case b | a we usually write F′q ⊂ Fq. Also,

Fq is a vector space over F′q of dimension b/a in that case.

Some facts concerning ideals in a (polynomial) ring R.

• The ideal I in R is a prime ideal if and only if R/I is a domain. Theideal I in R is a maximal ideal if and only if R/I is a field.

• If R = K[X] with K a field and X an indeterminate, then every idealI in R is of the form I = (f) for some polynomial f . Moreover, I is

1

2 Exercises

a prime ideal if and only if f is irreducible or f = 0. I is a maximalideal if and only if f is (non-constant and) irreducible.

• The morphism R → R/I, r 7→ r + I is called the natural map orcanonical map. (Here, I is an ideal in R.)

1 Which of the following polynomials is irreducible?

(a) X2 + X + 1 in F3[X].

(b) X3 + X + 1 in F4[X].

2 Which of the following quotient rings is a field?

(a) Z[X]/(X2 + 1),

(b) Q[X]/(X2 + 1),

(c) R[X]/(X2 + 1),

(d) C[X]/(X2 + 1).

Which ones of these sets do you know in a different guise as subsets of thecomplex numbers?

3 In the field F4 with 4 elements let α be an element 6= 0, 1.

(a) Show that α2 + α + 1 = 0. Find all zeros of X2 + X + 1 in F4.

(b) Show that the map F2[X] → F4, f(X) 7→ f(α) is a surjective morphismwith kernel (X2 + X + 1). Use this to prove that F2[X]/(X2 + X + 1)is isomorphic with F4.

4 Mark true or false.

(a) There exists a finite field with 36 elements.

(b) The group F∗9 contains an element of order 4.

(c) F2[X]/(X3 + X + 1) ∼= F2[X]/(X3 + X2 + 1).

5 An explicit copy of F25 is F5[X]/(X2 − 2). Let α be the class of X.

(a) Find a generator for F∗25, i.e., an element β ∈ F∗

25 such that F∗25 =

{β, β2, β3, . . . , β24}

1.1 Week 1 3

(b) 1 + α satisfies a quadratic equation with coefficients in F5. Find one.

6 Recall that every finite field of order pn (p prime) is isomorphic with a fieldof the form Fp[X]/(f), where f is an irreducible polynomial of degree n.

(a) Construct fields of orders 8 and 16, respectively, in the form F2[X]/(f).In both cases, find generators for the corresponding multiplicativegroups.

(b) Construct F16 in the form F4[X]/(f).

7 Let L be a field with 125 elements.

(a) Show that σ : x 7→ x5 is an automorphism (i.e., a bijective morphism)of L.

(b) What is the order of σ, i.e., the smallest positive integer n such thatσn is the identity?

(c) Show that Lσ = {x ∈ L|σ(x) = x} is a subfield of L. Which one?

8 The map σ : F16 → F16, x 7→ x4 is an automorphism of F16.

(a) Determine the order of σ.

(b) Determine the subfield Fσ16 = {x ∈ F16|σ(x) = x} of elements fixed

under σ. Write this subfield in the form F2(β) for some β and find theminimal polynomial h(X) ∈ F2[X] of β.

9 (Algebraist’s trick: constructing zeros) If K ⊂ L are two fields, thena polynomial f ∈ K[X] can also be viewed as a polynomial over L, i.e.,f ∈ L[X], since all the coefficients of f are in L.

Let K be a field and let f ∈ K[X] be an irreducible polynomial. (Thinkof X2 + 1 ∈ R[X] for example.)

• Verify that K can be viewed as a subset (a subfield) of the field L :=K[Y ]/(f(Y )) through the composition K ⊂ K[Y ] → K[Y ]/(f(Y )),k 7→ k + (f). Denote the class of Y by α.

• Now view f as a polynomial in L[X]. Show that α is a zero of f .

• Drop the assumption that f be irreducible, but suppose f is non-constant. Construct a suitable field L of the form K[Y ]/(g(Y )) suchthat f , viewed as polynomial in L[X], has at least one zero in L.

4 Exercises

10 Let L be a field and let K be a subfield of L, i.e., K is a field contained in L.Verify that L is a K–vectorspace. In each of the following cases determinethe K–vectorspace dimension.

(a) L = Q(i), K = Q.

(b) L = F2[X]/(X3 + X + 1), K = F2.

11 A partial order on a set S is a relation ≥ on S such that

(i) a ≥ a for every a ∈ S (the relation is reflexive);

(ii) if a ≥ b and b ≥ c then a ≥ c (the relation is transitive);

(iii) a ≥ b and b ≥ a imply a = b (the relation is antisymmetric).

A partial order is called a total order if, in addition,

(iv) for all a, b ∈ S, either a ≤ b or b ≤ a.

A total order is called a well–ordering if moreover the following holds:

(v) Every nonempty subset T of S contains a smallest element: there is at ∈ T such that t ≤ s for all t ∈ T .

A reduction order or monomial order on k[X1, . . . , Xn] is a well–ordering onthe set of monomials Xa such that

a > b and c ∈ Zn≥0 ⇒ a + c > b + c.

Verify that each of the following orders on the monomials is a reductionorder.

(a) The lexicographic order : a >lex b (or Xa >lex Xb) if the first nonzeroentry from the left in a − b is positive. We often abbreviate lexico-graphic order to ‘lex order’.

(b) The lexicographic total degree order or graded lex order (abbreviatedto ‘grlex’): a >grlex b (or Xa >grlex Xb) if deg Xa > deg Xb ordeg a = deg b and a >lex b. In words: graded lex order orders bytotal degree first and breaks ties using lex order.

12 Rewrite each of the following polynomials, ordering the terms using the lexorder, the grlex order and grevlex order. (Here, X1 > X2 > X3 > X4 andX > Y > Z.) Find lm(f), lt(f) and multideg(f) as well:

1.1 Week 1 5

(a) f(X, Y ) = 4X5Y − 9Y 8 + X3Y 3 + 8XY 4 − XY 7 + X2;

(b) f(X, Y, Z) = X − Y Z + 4X2 + X3 + XZ − 2Y ;

(c) f(X1, X2, X3, X4) = X21 + X2X

34 − 5X2

3X4 + 3X52 − 10X1X2X3.

13 The terms in each of the following polynomials occur in descending orderaccording to one of the monomial orders lex, grlex or grevlex. Find outwhich order in each case.

(a) f(X, Y, Z) = X4Y 5Z + 2X3Y 2Z − 4XY Z4;

(b) f(X, Y, Z) = 7X2Y 4Z + X2Y 2 − 2XY 6;

(c) f(X, Y, Z) = X2Z3 + XY 3Z + XY 2Z2.

14 Find the remainder on division of the polynomials below by the orderedset F = {XZ − Y 2, X3 − Z2} using first lex order and then grlex order.

(a) f(X, Y, Z) = −4X2Y 2Z2 + Y 6 + 3Z5;

(b) f(X, Y, Z) = XY − 5Z2 + X.

15 Let a1, a2, a3 be non-zero integers whose gcd is d. For every pair of indicesk, ` we define dk` = gcd(ak, a`). For each pair i < j we also define vij =(aj/dij) · ei − (ai/dij) · ej , where e1, e2, e3 denotes the standard basis. In thisexercise we consider integer solutions to the equation

a1x1 + a2x2 + a3x3 = 0

in the variables x1, x2, x3. We often write solutions in vector notation.

(a) How do you solve the equation a1x1 + a2x2 = 0 over the integers?

(b) Show that every solution of 4x + 6y + 9z = 0 is a Z-linear combina-tion of (3,−2, 0), (9, 0,−4) and (0, 3,−2). [Hint: compare the secondcoordinate of a solution with that of (3,−2, 0) + (0, 3,−2).]

(c) Let dk` = gcd(ak, a`). Show that d divides every dk`. Also show thatevery vij is a solution of a1x1 + a2x2 + a3x3 = 0.

(d) Let w = (w1, w2, w3) be a non-zero solution to the equation a1x1 +a2x2 + a3x3 = 0 and suppose w1 6= 0. Prove that w1 is a Z-linearcombination of a2/d12, a3/d13. [Hint: a1w1 = −a2w2 − a3w3 and notethat the gcd of a2/d, a3/d divides a1w1 and hence w1; also invoke theeuclidean algorithm.]

6 Exercises

(e) Show that the first coordinate of w minus a suitable combination ofthe v1k is 0. Use this observation to outline an algorithm for solvingthe equation a1x1 + a2x2 + a3x3 = 0 over the integers.

1.2 Week 2 7

1.2 Week 2

In some of the exercises, a reduction order is assumed without explicit men-tion. Polynomial rings are always polynomial rings over a field. In Mathe-matica various routines are available for Grobner basis related computations.For instance, Groebnerbasis[{poly1, . . .}, {x2, x1, x3}, MonomialOrder ->

DegreeLexicographic] computes a reduced Grobner basis (up to clear-ing denominators) using the graded lexorder with the variables ordered asindicated. Without the option a Grobner basis is computed w.r.t. the lex-order with the variables ordered as in the list of variables. Related routines:Eliminate, PolynomialReduce, PolynomialGCD, Solve (see Mathematica’sonline information for more details and more options).

1 Let f, g ∈ K[X1, . . . , Xn].

(a) Show that lt(fg) = lt(f) lt(g).

(b) Show that lm(f + g) ≤ max(lm(f), lm(g)). When does equality hold?

2 Compute the S-polynomial of each of the following pairs with respect tothe lexorder with X > Y > Z.

(a) X4 + 3X2 + 1 and 2X2 − X;

(b) X3 + X and −Y 2 + Y + 4;

(c) 3XY Z + X2 and X3 − Y 2 + 1.

Write a routine in Mathematica that produces the S-polynomial upon inputof a pair of polynomials.

3 Let f and g be non-zero polynomials.

(a) Let Xa and Xb be monomials. Express the S-polynomial of Xaf andXbg in terms of S(f, g).

(b) Let Xc be the least common multiple (lcm) of the leading monomialsof f and g. Show that the multidegree of S(f, g) is strictly smallerthan c.

4 In each case determine if the indicated set is a Grobner basis (with respectto the lexorder) for the ideal it generates.

(a) {X2 − 1};

8 Exercises

(b) {X + Y, X − Y } (with X > Y );

(c) {Y − X2, Z − X3} (with X > Y > Z);

(d) {Y − X2, Z − X3} (with Y > Z > X).

5 Explain what the Euclidean algorithm applied to the pair of polynomialsf, g ∈ K[X] and the Buchberger algorithm applied to the ideal (f, g) ⊂ K[X]have in common.

6 Let f = c1Xa1f1 + c2X

a2f2 be a sum whose multidegree is less than d,whereas each of the two terms has multidegree exactly d.

(a) Show that f is a multiple of S(f1, f2).

Extend this result to a sum of more than two terms: Let f =∑t

i=1 ciXaifi

be a polynomial with multidegree less than d. If ai + multideg(fi) = d forall i, then f can be written as a sum

f =∑

j,k

cj,kXd−γj,kS(fj , fk),

where Xγj,k is the least common multiple of lm(fj) and lm(fk).

(b) Let di = lc(fi). Use multideg(f) < d to prove that

t∑

i=1

cidi = 0.

(c) Set pi = Xaifi/di. Prove that

Xd−γj,kS(fj , fk) = pj − pk.

(d) We have f =∑t

i=1 cidipi. Use the trick of adding and subtractingterms and (b) to conclude that

f = c1d1Xd−c1,2S(g1, g2) + (c1d1 + c2d2)X

d−γ2,3S(g2, g3) + · · ·+(c1d1 + · · · + ct−1dt−1)X

d−γt−1,tS(gt−1, gt).

7 This exercise is about the relation between Gaussian elimination for solvinga system of linear equations and Buchberger’s algorithm for determiningGrobner bases. The reduction order we use is lexorder with x1 > x2 > · · · >xm. For an n×m matrix A = (aij) we let IA be the ideal generated by thelinear polynomials ak1x1 + · · · + akmxm corresponding to the rows of A.

1.2 Week 2 9

(a) Show that (the analogues of) elementary row operations applied tothe linear polynomials in IA leave IA unchanged. Conclude that ifA is changed into row reduced echelon form (=normaalvorm), thecorresponding linear polynomials still generate IA.

(b) Consider subtracting a row from another row in such a way that theleading term in the latter row is cancelled. Which S-polynomial isrelated to this row operation? What are the corresponding steps inGaussian elimination and the Buchberger algorithm?

(c) Show that the linear polynomials corresponding to the rows of a matrixA in row reduced echelon form are a Grobner basis for IA. [Hint:apply Reduce modulo this basis to S-polynomials of every pair of linearpolynomials; what variables are involved?]

8 Suppose the leading monomials of f, g ∈ K[X1, X2, . . . , Xn] are relativelyprime.

(a) Show that every term of S(f, g) is either a multiple of lt(f) or lt(g).

(b) Consider division of S(f, g) by f and g. Show that the divison algo-rithm does not stop right at the beginning. Show furthermore thatafter every step of the division algorithm the updated polynomial stillsatisfies the property that every term is either a multiple of lt(f) orof lt(g). Conclude that Reduce({f, g}, S(f, g)) = 0 and that {f, g} isa Grobner basis.

9 Let I = (f1, . . . , fm) be an ideal in the ring K[X1, . . . , Xn]. Let

Z(I) := {(x1, . . . , xn) ∈ Kn | f(x1, . . . , xn) = 0 ∀f ∈ I}

and

Z(f) = {(x1, . . . , xn) ∈ Kn | f(x1, . . . , xn) = 0}.

(a) Prove that Z(I) = Z(f1) ∩ Z(f2) ∩ · · · ∩ Z(fm).

(b) Provide a counterexample to the following statement: if f(x1, . . . , xn) =0 for all (x1, . . . , xn) ∈ Z(I), then f ∈ I. [Hint: try the one variablecase.]

10 Exercises

1.3 Week 3

Relevant facts. Rings are commutative with unit element.

• The ideal I in the ring R is called finitely generated (eindig voortge-bracht) if there exist elements a1, . . . , an ∈ I satisfying I = (a1, . . . , an),i.e. if I consists of all elements of the form r1a1 + · · · + rnan withr1, . . . , rn ∈ R.

• A ring R is called noetherian (noethers) if every ideal in R is finitelygenerated.

• An ascending chain of ideals (stijgende rij idealen) in the ring R is asequence of ideals I1, I2, . . . in R satisfying I1 ⊆ I2 ⊆ · · · .

• Lemma: The ring R is noetherian if and only if every ascending chainof ideals I1, I2, . . . in R stabilizes (i.e. there is an N such that IN =IN+1 = In+2 = · · · ).

• Hilbert’s Basis Theorem: R noetherian implies R[X] noetherian.

1 (Noetherian rings) Show that the following rings are noetherian.

(a) Z; Z[X]

(b) k, with k a field (lichaam); k[X], k[X, Y ].

2 (Noetherian rings) Let R en S be rings and let I be an ideal in R. Show:

(a) If R is noetherian and φ : R → S is a surjective morphism, then S isnoetherian.

(b) If R is noetherian then R/I is noetherian. In particular, quotients ofpolynomial rings over a field are noetherian.

3 (Noetherian rings) Let I1 ⊆ I2 ⊆ · · · be an ascending chain of ideals inthe ring R. de ring R.

(a) Show that the union ∪mIm is an ideaal in R.

(b) If the ring R is noetherian, then every chain of ideals in R stabilizes.Prove this.

(c) Prove the converse of (b): if every ascending chain of ideals in R stabi-lizes, then R is noetherian. [Hint: construct a chain (a1) ⊂ (a1, a2) ⊂· · · as far as needed.]

1.3 Week 3 11

4 (Hilbert’s Basis theorem:proof) Hilbert’s Basis Theorem states thatif R is a noetherian ring, then the polynomial ring R[X] is also noetherian.Here is a proof.

(a) Suppose the ideal I ⊂ R[X] is not finitely generated. Choose f1 ∈I \ {0} of minimal degree, say d1. Choose f2 ∈ I \ {f1} of minimaldegree, say d2. Why can this process be carried on indefinitely?

(b) Suppose ai is the leading coefficient of fi (i = 1, 2, . . .). What can yousay about the chain of ideals (a1), (a1, a2), . . . in R?

(c) Suppose ak+1 = b1a1 + · · · + bkak. Use the polynomial

fk+1 −k

∑

i=1

bifiXdk+1−di

to arrive at a contradiction.

5 (On total orders) Let < be a total order on a set S. Show that every non-empty set of S has a minimal element if and only if every infinite sequences1 > s2 > s3 > · · · eventually stabilizes, i.e., satisfies sN = sN+1 = sN+2 =. . . for some N .

6 (On monomial orders)

(a) Show that on the set of monomials in one variable X the only monomialordering is X < X2 < X3 < . . ..

(b) Consider K[X1, . . . , Xn] with the lexorder, where X1 > X2 > · · · .Show that for i = 1, . . . , n − 1, the monomial Xi is greater than anymonomial in the variables Xi+1, Xi+2, . . .. Suppose the leading term ofthe polynomial f contains only powers of the variables X`, X`+1, . . ..Show that the same holds for every term of f .

7 (New monomial orders) Let < be a monomial ordering on the set M ofmonomials of the ring K[X1, . . . , Xn]. Let B be a finite totally ordered set;the order on B is also denoted by <. We define the relation <G on the set Gof maps B → M as follows: g <G g′ if the largest element b ∈ B satisfyingg(b) 6= g′(b) actually satisfies g(b) < g′(b).

(a) Take n = 1 (then the only possibility for the ordering on M is thelexorder) and let B = {1, 2, . . . , m} with the natural ordering. Identify

12 Exercises

each map a : B → M with a monomial in m indeterminates in theobvious way. Show that <G coincides with a lexorder on the monomialsin K[X1, . . . , Xm].

(b) Show that <G is a monomial ordering [Hint: to show that every de-scending sequence g1 >G g2 >G g3 >G · · · must stabilize, first considerthe sequence g1(b), g2(b), . . ., where b is the largest element of B.]

(c) Can you find a relation with a lexorder (in many more variables) likein (a)?

8 Let < be a monomial ordering on the set M of monomials of the ringK[X1, . . . , Xn]. Consider the set

H = {f : M → N | f−1({1, 2, . . .}) is finite},

i.e., the collection of maps with nonzero image for only finitely many ele-ments of M. The relation <H on H is defined as follows: h <H h′ if and onlyif the smallest element m ∈ M with h(m) 6= h′(m) satisfies h(m) < h′(m).Show that < H is well-ordered.

9 (System of equations) Solve the following system of equations usingGrobner bases (and Mathematica):

x2 + y2 + z2 = 3,x + y + z = 2,2x + y = 2.

Experiment with various monomial orders in solving the system.

10 [On monomial ideals] A monomial ideal in K[X1, . . . , Xn] is an ideal thatcan be generated by monomials. Let I be a monomial ideal generated bythe monomials Xa, where a runs through some index set A.

(a) Let f ∈ K[X1, . . . , Xn]. Prove: f ∈ I ⇔ every term/monomial of f isin I.

(b) Show: Xb ∈ I ⇔ Xa | Xb for some a ∈ A. Observe that thisstatement has the following implication for ideals generated by lead-ing terms of polynomials: if lt(f) ∈ (lt(g1), . . . , lt(gt)) then lt(f) isdivisible by at least one of the lt(g1), . . . , lt(gt).

(c) Use Hilbert’s basis theorem to show that there exist a(1), . . . ,a(m) ∈A with I = (Xa(1), . . . , Xa(m)).

1.3 Week 3 13

11 Let B = {b1, . . . , bt} be a Grobner basis for the ideal I ⊂ K[X1, . . . , Xn]and let f ∈ K[X1, . . . , Xn]. The algorithm Reduce produces an element rsuch that

• no term of r is divisible by any of the leading terms lt(b1), . . . , lt(bt),

• there is a g ∈ I with f = g + r.

(Verify this!)

(a) Show that r is the unique element satisfying both conditions (so divi-sion with remainder leaves a unique remainder in this setting). [Hint:use part (b) of Exercise 10.]

(b) Show that f ∈ I if and only Reduce(B, f) = 0. Show how this obser-vation together with the Buchberger algorithm solves the ideal mem-bership problem: how to decide if a given element belongs to an ideal.

(c) Give a few alternatives for proving that two ideals are equal.

12 Find a non-trivial equation in x, y, z whose solutions contain the surfacegiven by the following parametric representation:

x = t + u, y = t2 + 2tu, z = t3 + 3t2u.

13 A Grobner basis B is called minimal if

• lc(b) = 1 for every b ∈ B;

• no leading term lt(b) of b ∈ B is contained in the ideal generated bythe leading terms of B \ {b}.

(a) If B is a Grobner basis and b ∈ B satisfies lt(b) ∈ (lt(B \ {b})), thenB \ {b} is a Grobner basis (for the same ideal). Prove this.

(b) Show how to construct a minimal Grobner basis out of a given one.

(c) Let B and B′ be minimal Grobner bases for the same ideal. Provethat the sets of leading terms coincide: lt(B) = lt(B ′). In particular,the cardinalities of lt(B) and lt(B ′) are the same.

14 Exercises

1.4 Week 4

1 Prove each of the following statements.

(a) In this part > denotes lexorder with X1 > X2 > · · · > Xn. Supposef ∈ k[X1, . . . , Xn] satisfies lt(f) ∈ k[X`+1, . . . , Xn]. Show that f ∈k[X`+1, . . . , Xn].

(b) If B is a Grobner basis for the ideal I and lt(b) ∈ (lt(B \{b})) for someb ∈ B, then B \ {b} is a Grobner basis of I.

(c) If B and B′ are Grobner bases of the ideal I, then the union B ∪B ′ isa Grobner basis of I.

(d) Let G = {g1, . . . , gt} be a Grobner basis of the ideal I. Show thatf ∈ I if and only if f leaves remainder 0 upon division by G.

2 A Grobner basis B of the ideal I is reduced if every element in B hasleading coefficient 1 and if the following holds for every b ∈ B: no termof b is divisible by any of the leading terms of the b′ ∈ B with b′ 6= b. Inparticular, reduced Grobner bases are minimal.

(a) Let B be a minimal Grobner basis. For b ∈ B, replace b by b′ :=Reduce(B \ {b}, b). Show that lt(b) = lt(b′) and that no term of b′ isdivisible by any of the leading terms of the elements of B \ {b}.

(b) Use (a) to construct a reduced Grobner basis from a minimal one.

(c) Prove: if B and B′ are both reduced Grobner bases for the ideal I,then B = B′. [Hint: First use minimality to conclude lt(B) = lt(B ′).Suppose lt(b) = lt(b′) for some b ∈ B and b′ ∈ B′. Consider divisionof b − b′ by B and conclude that b = b′.]

3 Let I ⊂ K[X1, . . . , Xn] be an ideal. The set of common zeros is denoted byZ(I) (see Week 2, Exercise 9). The set

√I is defined as {f ∈ K[X1, . . . , Xn] |

fm ∈ I for some positive integer m}.

(a) Prove that√

I is an ideal. This ideal is called the radical of I (Dutch:wortelideaal van I).

(b) If f ∈√

I then f(x1, . . . , xn) = 0 for every (x1, . . . , xn) ∈ Z(I). Provethis.

1.4 Week 4 15

(c) Give an example to show that the converse is false: if f(x1, . . . , xn) = 0for every (x1, . . . , xn) ∈ Z(I) then f ∈

√I.

(d) Show that the converse is true if K = C and n = 1. (The so-calledNullstellensatz states that in general the converse is true when K isalgebraically closed.)

4 Suppose R is a domain and f ∈ R \ {0}. The subset

R[1

f] = {r0 +

r1

f+

r2

f2+ · · · + rm

fm| m ∈ N, r0, . . . , rm ∈ R}

of the field of quotients Q(R) of R is actually a subring of Q(R). It is thesmallest subring of Q(R) containing R and 1/f .

(a) Verify that in any ring S the identity

1 − sm = (1 − s)(1 + s + s2 + · · · + sm−1)

holds for every positive integer m and every s ∈ S.

(b) Define the map φ : R[X] → R[1/f ] by

r0 + r1X + · · · + rmXm 7→ r0 +r1

f+

r2

f2+ · · · + rm

fm.

Prove that the kernel of this map is (1−Xf). Deduce that R[X]/(1−Xf) is isomorphic with R[1/f ]. So computing with the class of X islike computing with the multiplicative inverse of f .

5 Let I ⊂ K[X1, . . . , Xn] be an ideal generated by f1, . . . , fr and let f ∈K[X1, . . . , Xn]. Let T be a new indeterminate and consider the ideal J inK[X1, . . . , Xn, T ] generated by I and 1 − Tf .

(a) Suppose 1 ∈ J so that 1 = q1f1 + · · · + qrfr + q(1 − Tf) for certainq1, . . . , qr, q ∈ K[X1, . . . , Xn, T ]. Make a suitable substitution for T inthis relation to show that fm ∈ I for some positive integer m.

(b) Suppose fm ∈ I for some positive integer m. Use the identity 1 =(1 − fmTm) + fmTm to show that 1 ∈ J .

(c) Use the previous items and Grobner bases to solve the radical mem-bership problem: when is f ∈

√I? (The radical

√I was defined in

Exercise 3.)

16 Exercises

6 Let I be an ideal in the polynomial ring R = k[X1, . . . , Xn] with I 6= Rand let a ∈ R. The image of a in R/I is denoted by a.

a) Show that a is a unit in R/I if and only if R = (a) + I.

b) If I = (f1, . . . , fk), how can Grobner bases be used to check that a hasa multiplicative inverse in R/I?

c) Suppose a is a unit in R/I. Use the ideal I+(aY −1) in k[X1, . . . , Xn, Y ]to determine an inverse for a.

d) Let I = (X2−Y 2−1, Y −X2−Y 2) ⊂ Q [X, Y ]. Determine the inverseof Y in Q [X, Y ]/I. Give more examples.

7 Let α ∈ C. Then α is called algebraic (over Q) if there is a nonconstantpolynomial f(X) ∈ Q [X] such that f(α) = 0. It is a fact that the algebraicnumbers form a subfield of C.

(a) Show that every rational number is algebraic.

(b) Show that every one of the following complex numbers is algebraic:√2, e2iπ/3,

√2 + i. [Hint for the last one: let α =

√2 + i and square

α −√

2.]

(c) Let α be an algebraic number and let p ∈ Q[X] be a non-constantpolynomial of minimal degree satisfying p(α) = 0. Prove that p isirreducible and that every polynomial q ∈ Q[X] satisfying q(α) = 0is a multiple of p. [Hint: division with remainder.] If the leadingcoefficient of p is taken to be 1, then p is called the minimal polynomial(over Q) of α.

(d) Given an algebraic number α ∈ C, the field Q (α) consists of all ex-pressions of the form

g(α)

h(α),

where g, h are polynomials and where h(α) 6= 0. It is the smallestsubfield of C containing α. Define the map F : Q [X] → Q(α) byq(X) 7→ q(α). Show that the kernel of this map is the ideal generatedby the minimal polynomial of α. Prove that Q [X]/(p) is isomorphicwith Q(α). [Hint: rings modulo maximal ideals are fields.] Concludealso that Q(α) = Q [α] and that Q(α) is an n-dimensional Q-vectorspace with (Q-)basis 1, α, α2, . . . , αn−1.

1.4 Week 4 17

8 Let α be an algebraic number with minimal polynomial p(X) ∈ Q [X]. ByExercise 7 the abstract way to compute with α is to compute modulo p(X),i.e., to compute in Q [X]/(p(X)). Suppose we want to compute the minimalpolynomial of some polynomial expression g(α).

(a) Explain why the class of Y in Q [X, Y ]/(p(X), Y −g(X)) is an abstractway of representing g(α).

(b) What kind of Grobner basis computation and, in particular, whatkind of monomial ordering, would you use to produce the minimalpolynomial of α? [Hint: elimination of variables.]

(c) Find the minimal polynomial of ( 3√

2)2 − 3√

2.

9 Let α and β be algebraic numbers. Suppose α has minimal polynomialp(X) and β has minimal polynomial q(X). We want to compute the minimalpolynomial of some polynomial expression f(α, β) of α and β. (Since thealgebraic numbers form a subfield of C, such expressions are again algebraic.)By Exercise 7 the abstract way to compute with α is to compute modulop(X). Similarly, computing with β comes down to computing modulo q(X).To separate these two, we work in Q [X, Y ]/(p(X), q(Y )). To compute withf(α, β) we introduce a third variable Z and add the generator Z − f(X, Y )to the ideal:

Q [X, Y, Z]/(p(X), q(Y ), Z − f(X, Y )).

(a) Show why it is plausible that the class of Z represents f(α, β).

(b) What kind of Grobner basis computation (e.g., what kind of mono-mial ordering) would you use to produce a candidate for the minimalpolynomial of f(α, β)? (There is an exact statement under suitableconditions, but we refrain from stating and proving it here.)

(c) Apply the previous item to compute the minimal polynomials of thefollowing numbers:

√2 + 3

√2, i +

√2, 3

√5 +

√7 +

√11.

10 Let A, B, C, D be points in the plane. Express each of the following state-ments about these points in terms of polynomial equations.

(a) A and B and C are equidistant.

(b) AB is perpendicular to CD.

(c) AB is parallel to CD.

18 Exercises

11 Consider a right triangle ABC, whose angle at A is 90◦. The foot of thealtitude from A on BC is called H. The statement is that H and themidpoints P (of AB), Q (of BC), R (of AC) of the three sides lie on a circle(this is known as the Circle theorem of Apollonius).

A B

C H

P

QR

Before we go into a proof using Grobner bases, we sketch the classicalapproach. The vertex A and the three midpoints P, Q, R form a rectan-gle and lie on the circle with diameters AQ and PR (AQ and PR passthroughthe center of the circle and the angles ∠APQ and ∠ARQ are rightangles). Since the triangle AHQ has a right angle at H, the point H is alsoon thecircle with diameter AQ.

Now we turn to the description in terms of polynomials. We choosecoordinates in such a way that A is at the origin, that B = (2x, 0) andC = (0, 2y). Then P = (x, 0), Q = (x, y) and R = (0, y).

(a) The point H = (p, q) is on BC and AH is perpendicular to BC.Express these two conditions as polynomial equations. Let I be theideal generated by the two polynomials.

(b) What is the equation of the circle passing through P, Q, R (in termsof the variables x, y and, if needed, new variables for the center andradius)? What is the ideal membership problem corresponding to thetheorem to be proved? Use a Grobner basis computation to answerthe latter question in the negative! Show that y times your polynomialdoes belong to the ideal.

(c) The phenomenon you just encountered is related to the fact that in aclassical geometry statement often implicit assumptions are made (e.g.,the triangle is non-degenerate) that are necessary for the validity of theresult. In the case at hand such a non-degeneracy condition is y 6= 0.To deal with this extra condition algebraically, one introduces a new

1.4 Week 4 19

indeterminate T and considers the ideal (in the extended polynomialring) generated by the generators so far and by 1−yT . Solve the idealmembership problem for y times your polynomial with respect to thenew ideal.

12 (The intersection of ideals) Let I and J be ideals in the polynomialring K[X1, X2, . . . , Xn]. It follows from Algebra 2 that the intersectionI ∩ J and the sum I + J are again ideals in K[X1, X2, . . . , Xn]. SupposeI = (f1, f2, . . . , fr) and J = (g1, g2, . . . , gs).

(a) How would you compute a Grobner basis of I + J (with respect toa given monomial ordering)? If n = 1 how would you compute aGrobner basis of I ∩ J?

(b) Let T be a new indeterminate. The sets TI = {Tf | f ∈ I} and(1 − T )J = {(1 − T )g | g ∈ J} generate ideals (TI) and ((1 − T )J)in K[X1, X2, . . . , Xn, T ]. Show that (TI) is generated by Tf1, . . . , T fr

and that (1 − T )I is generated by (1 − T )g1, . . . , (1 − T )gs. Show bygiving a counterexample that I and J are not necessarily ideals inK[X1, X2, . . . , Xn, T ].

(c) Prove that

I ∩ J = (TI + (1 − T )J) ∩ K[X1, X2, . . . , Xn].

[Hint for ⊃: if f(X1, . . . , Xn) = g(X1, . . . , Xn, T ) + h(X1, . . . , Xn, T )with g(X1, . . . , Xn, T ) ∈ TI and h(X1, . . . , Xn, T ) ∈ (1 − T )J , thensubstitute T = 0 and T = 1, respectively.]

(d) Show how the previous item together with Proposition 5.5 (Elimina-tion of variables) leads to an algorithm for computing the intersectionof ideals. Experiment with a few examples of your own choice.

(e) In the case the ideals I = (f) and J = (g) are both generated by one el-ement, show that I ∩J = (lcm(f, g)). Conclude that the algorithm forcomputing intersections enables you to compute least common mul-tiples (see Appendix, ??) without factoring polynomials first. Alsoshow how to compute gcd’s without factoring. (Note that there isno analogue of the euclidean algorithm in the case of more than onevariable.)

20 Exercises

13 Let f(X1, . . . , Xn) be a polynomial in K[X1, . . . , Xn]. We say that f issymmetric it it remains unchanged under any permutation of the variablesX1, . . . , Xn:

f(Xσ(1), . . . , Xσ(n)) = f(X1, . . . , Xn) for all σ ∈ Sn.

(a) Verify that X1X2 + X1X3 + X2X3 and X21 + X2

2 + X23 are symmetric

polynomials in three variables.

The elementary symmetric polynomials in n variables are the polynomials

σr(X1, . . . , Xn) =∑

i1<i2<···<ir

Xi1Xi2 · · ·Xir r = 1, . . . , n.

(b) Write down the elementary symmetric polynomials σ1, σ2, σ3 in threevariables.

(c) The identity X21 +X2

2 = (X1 +X2)2−2X1X2 = σ2

1 −2σ2 expresses thesymmetric polynomial X2

1 +X22 in terms of the elementary symmetric

polynomials σ1, σ2 (in two variables). Show that X31 + X3

2 can beexpressed in terms of elementary symmetric polynomials as well.

(d) Verify that, upon fully expanding, the coefficients of (X − α1)(X −α2) · · · (X −αn) are (up to a sign) elementary symmetric polynomialsin the αi.

Suppose f ∈ K[X1, . . . , Xn] is a (non-zero) symmetric polynomial. Fix thelexorder with X1 > X2 > · · · > Xn.

(e) Show that if αXa is a term of f then αXb is a term of f for anyvector b whose coordinates are a permutation of those of a. Concludethat if αXa is the leading term of f , where a = (a1, . . . , an), thena1 ≥ a2 ≥ · · · ≥ an.

(f) Suppose αXa, with a = (a1, . . . , an), is the leading term of f . Showthat the leading term of α σa1−a2

1 σa2−a3

2 · · ·σan−1−an

n−1 σann is αXa. Use

this observation to establish that every symmetric polynomial can beexpressed as a polynomial in the elementary symmetric polynomials.(In fact, in a unique way.) Write an algorithm that takes as input asymmetric polynomial f and outputs f expressed in terms of elemen-tary symmetric polynomials. Try it out on various polynomials, suchas (X2 + Y 2)(X2 + Z2)(Y 2 + Z2).

1.4 Week 4 21

(g) There is also a Grobner-basis approach to the problem of express-ing a symmetric polynomial in terms of elementary ones (we willnot ask you to prove it here). Let Y1, . . . , Yn be new indeterminatesand let B be a Grobner-basis for the ideal (σ1 − Y1, . . . , σn − Yn) ⊂K[X1, . . . , Xn, Y1, . . . , Yn] with respect to the lexorder with X1 > X2 >· · · > Xn > Y1 > Y2 > · · · > Yn. Let f ∈ K[X1, . . . , Xn] and let g bethe remainder of f on division by B. Then

(i) f is symmetric if and only if g ∈ K[Y1, . . . , Yn];

(ii) If f is symmetric then f = g(σ1, . . . , σn) expresses f (uniquely)in terms of the elementary symmetric polynomials.

Try out this approach.

14 Let A, B be k × k matrices over R and consider the linear maps A,B :Rk → Rk defined by Ax = A x> and Bx = B x>. In this exercise we willrelate the kernels of A and B. Suppose there exist k × k matrices F and Gsuch that A F> = B and B G> = A.

(a) Show that Ax = B(x G).

(b) Let H be a matrix such that the rows of H form a basis of ker(B).What is the size of H? If x ∈ ker(A), show that there exists a y ∈ Rk

such that x G = y H.

(c) Use the equality x = x(I − GF ) + xGF to prove that every vector inthe kernel of A is a linear combination of the rows of I −GF and HF .

(d) Verify that A(I − GF )> = 0 and A(HF )> = 0 and conclude that thekernel of A is precisely the space spanned by the rows of I − GF andHF .

(e) If A and B are changed into `×k matrices over a field, would a similarresult still hold? (Of course, the sizes of F and G also change.)

15 Find (generators for) all syzygies of (XY 3−1, X3Y −1). Use, for example,the following steps.

(a) Compute a (reduced) Grobner basis for the ideal generated by XY 3−1and X3Y − 1 with respect to the lexorder with X > Y . Do this byhand and keep track of how the new basis is expressed in the old one.

22 Exercises

(b) Express XY 3 − 1, X3Y − 1 in terms of the members of the Grobnerbasis, and, conversely, express the members of the Grobner basis interms of XY 3 − 1, X3Y − 1.

(c) Find the syzygies for the Grobner basis.

16 Let a1, . . . , an be non-zero integers whose gcd is d. For every pair of indicesk, ` we define dk` = gcd(ak, a`). For each pair i < j we also define vij =aj/dijei − ai/dijej , where e1, . . . , en is the standard basis. In this exercisewe consider integer solutions to the equation

a1x1 + a2x2 + · · · + anxn = 0

in the variables x1, . . . , xn.

(a) Show that every solution of 4x + 6y + 9z = 0 is a Z-linear combina-tion of (3,−2, 0), (9, 0,−4) and (0, 3,−2). [Hint: compare the secondcoordinate of a solution with that of (3,−2, 0) + (0, 3,−2).]

(b) Let dk` = gcd(ak, a`). Show that d divides every dk`. Also show thatevery vij is a solution of a1x1 + a2x2 + · · · + anxn = 0.

(c) Let w = (w1, . . . , wn) be a non-zero solution to the equation a1x1 +a2x2 + · · ·+anxn = 0 and suppose w1 6= 0. Prove that w1 is a Z-linearcombination of a2/d12, a3/d13, . . . , an/d1n. [Hint: a1w1 = −a2w2 −a3w3 − · · · − anwn and note that the gcd of a2/d, . . . , an/d dividesa1w1 and hence w1; also invoke the euclidean algorithm.]

(d) Show that the first coordinate of w minus a suitable combination ofthe v1k is 0. Use this observation to outline an algorithm for solvingthe equation a1x1 + a2x2 + · · · + anxn = 0 over the integers.

17 Consider the triangle 4ABC. How can we translate the geometric resultsbelow as a system of polynomial equations? Write down the hypothesis andthe conclusion.

(a) Draw the three altitudes. Then these three segments intersect at onepoint called the orthocenter of the triangle.

(b) Say A1 is the midpoint of BC, B1 is the midpoint of AC and C1 isthe midpoint of AB. Then the segments AA1, BB1 and CC1 meet atone point called the centroid of the triangle.

1.4 Week 4 23

18 Let I be the ideal in C[x, y, z] determined by the system of equations:

x5 +1

x5= y

x +1

x= z

(a) Find basis for the elimination ideals I1 and I2.

(b) Verify that you can use z as a parameter, that is, that you can writeformulas for x and y in terms of z.

19 Same questions as in the previous problem for the system of equations:

x2 + y2 + z2 = 1xyz = 1

20 Let us prove the Weak Nullstellensatz Theorem: Let K be an algebraicclosed field. Let I = 〈f1, . . . , fs〉 be an ideal in K [x1, . . . , xn]. If V(I) = ∅,then I = K[x1, . . . , xn].

We will prove it by induction.

(a) Using the fact that K [x] is a principal ideal domain, prove the state-ment for n = 1.

(b) Suppose that f has total degree N ≥ 1. Do some change of variablesto check that we can assume that for some constant c 6= 0:

f1(x1, . . . , xn) = cxN1 + terms with lower degree in x1

(c) We will use now the following result: Let I = 〈f1, . . . , fs〉 ⊂ K[x1, . . . , xn].Assume that for some i:

fi(x1, . . . , xn) = cxN1 + terms with lower degree of x1

where c ∈ C∗ and N > 0. If I1 is the first elimination ideal, then

π(V(I)) = V(I1)

where π is the projection on the last n − 1 factors.

Use this result and the induction hypothesis to finish the Weak Null-stellensatz proof.

24 Exercises

21 In the proof of Theorem ?? we have the following statements:

Q(α)[Y ]/(q(Y )) ∼= Q(α, β) and Q(α) ∼= Q[X]/(p(X))

How can you use these facts to conclude that

Q[X, Y ]/(p(X), q(Y )) ∼= Q(α, β)?

The following steps may be useful.

(a) Show that the map Q [X, Y ] → Q (α)[Y ], f(X, Y ) 7→ f(α, Y ) haskernel (p(X)), i.e., the ideal generated by p(X). Conclude that

Q [X, Y ]/(p(X)) ∼= Q (α)[Y ].

(b) [Third Isomorphism Theorem] If I and J are ideals in the ring R,satisfying I ⊂ J , then the morphism R/I → R/J , a + I 7→ a + J issurjective with kernel J/I. Prove this and conclude that there is an(natural) isomorphism

(R/I)/(J/I) ∼= R/J,

and apply this to the ideals (p(X)) ⊂ (p(X), q(Y )).

22 Find the minimal polynomials over Q of the following complex numbers.

(a) i +√

2.

(b)( 3√

5 +√

7 +√

11)

2.

(c)(i√

3 − 3√

2)

2.

23 Show that if α has minimum polynomial t2−2 over Q and β has minimumpolynomial t2 − 4t + 2 over Q, then the extensions Q(α) : Q and Q(β) : Qare isomorphic.

24 Find the degree of the following extensions.

(a) Q(α) : Q where α is the real cube root of 2.

(b) Q(i,√

6) : Q.

1.4 Week 4 25

(c) Q(ρ, 3√

2) : Q where ρ3 = 1 and ρ 6= 1.

(d) Q(√

(1 +√

3) : Q).

(e) Q(√

2,√

3,√

5) : Q.

25 Prove that Q(√

3,√

5) = Q(√

3 +√

5).

26 Exercises

1.5 Week 5

1 Let I be an ideal in the ring R = k[X1, . . . , Xn].

(a) Let f ∈ R. Show that f + I ∈ R/I has an inverse in R/I if and onlyif 1 ∈ I + (f) (the ideal generated by I and f). How would you verifythis using Grobner bases?

(b) Let I = (f1, . . . , fr) and let G = {g1, . . . , gt} be a Grobner basis forI. Show that with extra bookkeeping the Grobner basis algorithm canalso output a t × r matrix A with polynomial entries such that

g1...gt

= A ·

f1...fr

.

(c) Suppose f + I ∈ R/I has an inverse. Describe a method to computethe inverse. Apply the method in a quotient ring of your choice.

2 (Content of a polynomial) Let f and g be non-zero polynomials in Z[X].

(a) Suppose r ∈ Q is such that rf has integral coefficients and c(rf) =c(f). Show that r = ±1.

(b) Suppose f = gh with g ∈ Z[X]. Prove that there exists an integer dsuch that g/d ∈ Z[X] and dh ∈ Z[X].

(c) Let h ∈ Q[X] be a common divisor of f and g. Show that there existsa rational number r such that

f = rh · h1 and g = rh · h2,

with h1, h2 ∈ Z[X]. [Hint: reduce to the case of the previous item.]

3 (Eisenstein’s criterion) Let p be a prime and let

f = amXm + am−1Xm−1 + · · · + a1X + a0

be a polynomial with integer coefficients satisfying

• p does not divide am;

• p divides am−1, am−2, . . . , a0;

1.5 Week 5 27

• p2 does not divide a0.

(a) Prove that f is irreducible in Q [X]. This result is known as Eisen-stein’s criterion. [Hint: assume f factors as f = gh with g and h bothin Z [X] and of positive degree, and reduce mod p.]

(b) Apply Eisenstein’s criterion to show that the polynomial X4+5X +10is irreducible over Q[X].

(c) Show by example that none of the conditions on f can be dropped.

4 (Irreducibility criteria) Let K be a field and f ∈ K[X].

(a) Let a ∈ K. Show that f is irreducible if and only if f(X + a) isirreducible. Find out what the effect of the transformation is if a is azero of f .

(b) Suppose f has degree n ≥ 1 and has non-zero constant term. Provethat f is irreducible if and only if Xn f(1/X) is irreducible. Applythis result and Eisenstein’s criterion to show that 3X4 + 6X + 1 isirreducible over Q[X].

5 Use any irreducibility criterion you know (Eisenstein’s criterion, reductionmod p, etc.) in order to decide whether the following polynomials are irre-ducible or not.

(a) 29X5 + 5

3X4 + X3 + 13 over Q[X];

(b) X4 + X3 + X2 + X + 1 over Q[X] [Hint: replace X by X + 1];

(c) X4 + 15X3 + 7 over Z[X];

(d) X7 + 11X3 + 33X + 22 over Q[X];

(f) X3 − 7X2 + 3X + 3 over Q[X];

28 Exercises

1.6 Week 6

1 (Polynomial identities) Every element x in the finite field Fq satisfiesxq = x. This exercise is about related identities for polynomials in Fq[X].

(a) Show that every f ∈ Fq[X] satisfies

f(X)q = f(Xq).

In particular, if all exponents of a polynomial in Fq[X] are divisible byq, then the polynomial is a q-th power. For example, X8 + X4 + 1 ∈F2[X] equals (X2 + X + 1)4.

(b) Prove that Xq −X = Πa∈Fq(X − a) implies f q − f = Πa∈Fq(f − a) forevery f ∈ Fq[X].

(c) Show that F∗q consists of (q−1)/2 squares (S) and (q−1)/2 non-squares

(N). Prove that the squares (resp. non-squares) are zeros of

X(q−1)/2 − 1 resp. X(q−1)/2 + 1.

(d) Suppose q is odd and the polynomials f and v satisfy f |vq−1 − 1.Let f = f e1

1 · · · f err be the factorisation of f in irreducibles and let

si ∈ Fq (i = 1, . . . , r) satisfy v ≡ si (mod f eii ). Show that f divides

v(q−1)/2 − 1 or v(q−1)/2 + 1 if and only if either all si are squares or allsi are non-squares.

2 (Berlekamp’s method) In each of the following cases find the factoriza-tion of the polynomial f using Berlekamp’s method.

(a) Let f = X3 + X2 − X − 1 ∈ F3[X]. (First determine a v of the formaX2 + bX such that v3 ≡ v (mod f).)

(b) f = X4 + X3 + X + 1 ∈ F2[X].

3 (Chinese remainder theorem) The rings Z/12Z, Z/3Z and Z/4Z arerelated by the isomorphism

Z/12Z → Z/3Z × Z/4Z, a + 12Z 7→ (a + 3Z, a + 4Z).

This is a particular instance and formulation of the Chinese RemainderTheorem for integers (Algebra 1). This exercise is about a similar statementin the context of polynomials.

Suppose f ∈ Fq[X] is the product of two factors g and h, which arerelatively prime.

1.6 Week 6 29

(a) Consider the map

φ : Fq[X] → Fq[X]/(g) × Fq[X]/(h), r 7→ (r + (g), r + (h)).

Show that this map is a morphism with kernel (f).

(b) Show that φ induces a morphism

φ : Fq[X]/(f) → Fq[X]/(g) × Fq[X]/(h)

and argue that this morphism is in fact an isomorphism. (Why is itinjective? Why is it surjective?)

(c) Generalize to the case that f is the product of 2 or more relativelyprime factors (no proof asked).

(d) Let f1, . . . , fs be relatively prime polynomials, and let g1, . . . , gs beany polynomials. Show that there exists a polynomial a satisfying thesystem of equations

a ≡ g1 (mod f1), . . . , a ≡ gs (mod fs).

[Hint: apply induction; the case s = 2 is the crucial case.]

Also show that any two solutions differ by a multiple of the productf1f2 · · · fs.

4 Let f ∈ Fq[X] be a non-trivial polynomial.

(a) Show that

V = {v ∈ Fq[X] | vq ≡ v (mod f)},W = {v ∈ Fq[X]/(f) | vq = v}

are both Fq vector spaces. Why is V infinite dimensional?

(b) Determine W if f is irreducible.

(c) Suppose q = pm with p prime and f = f e1

1 · · · f err , where the fi are

distinct irreducible factors. Consider

U = {v ∈ Fq[X]/(f) | vp = v}.Show that under the isomorphism

Fq[X]/(f) ∼= Fq[X]/(f e1

1 ) × · · · × Fq[X]/(f err )

U corresponds precisely to the subset Frp of the right-hand side. In

particular, U is an r-dimensional vector space over Fp (cf. Tapas, Ch.4, Th. 2.2).

30 Exercises

5 (Derivatives and squarefree polynomials) Given a polynomial f =a0 + a1X + · · · + amXm with coefficients in a field K, we define the for-mal derivative Df or f ′ of f as the polynomial

a1 + 2a2X + 3a3X2 + · · · + mamXm−1.

It is not hard to verify that the formal derivative satisfies the usual rules fordifferentiation

(f + g)′ = f ′ + g′, (fg)′ = f ′g + fg′.

(The same definition works over a ring, but we will not consider that case.)

(a) What is the degree of the derivative in terms of that of f? Distinguishbetween the cases char(K) = 0 and char(K) 6= 0.

(b) Suppose f is divisible by g2 for some polynomial g. Show that f andDf have g as a common factor. In particular, if gcd(f, Df) = 1, thenf is squarefree, i.e., f is the product of distinct irreducible factors.

(c) Conclude from the previous item that if f and Df are relatively prime,then f cannot have a zero of multiplicity at least 2 in any extensionof K. In particular, if f is irreducible and char(K) = 0 then f has nomultiple roots in any extension.

(d) Suppose K has characteristic 0. Prove that f and Df are relativelyprime, if f is not divisible by the square of any non-trivial polynomial.[Hint: suppose g is an irreducible common factor of f and Df andconsider D(gh), where f = gh.]

(e) Suppose K = Fp, where p is a prime. Let f be a polynomial of positivedegree such that Df = 0. Show that f = gp for some polynomial f .(See Exercise 1.)

6 (Factoring and linear algebra) Let f ∈ Fq[X] be a polynomial of de-gree n > 0. In this exercise we show how to reduce solving the equationvq ≡ v (mod f) to linear algebra. An Fq vector space basis of Fq[X]/(f)is given by 1, X, . . . , Xn−1. Let A be the n × n-matrix whose i-th column(i = 1, . . . , n) contains the coefficients with respect to this basis of theunique representative of X (i−1)q (mod f) of degree less than n. For everyv = a0 + a1X + · · · + an−1X

n−1 ∈ Fq[X] of degree less than n write v forthe corresponding coefficient vector (a0, a1, . . . , an−1) ∈ Fn

q .

1.6 Week 6 31

(a) Show that the map

Fq[X]/(f) → Fq[X]/(f), v 7→ vq − v.

is an Fq-linear map whose matrix with respect to the indicated basisis A.

(b) Show that the coefficient vector vq of vq, where deg(v) < n, equals thematrix product A · v>. What ‘vector’ do you have to multiply A · v>with to find the unique representative of vq mod f of degree less thann?

(c) Use the previous items to show that solving vq ≡ v (mod f) comesdown to solving the linear system of equations with coefficient matrixA − I.

(d) Find a basis for this kernel if f = X12 − 1 ∈ F5[X].

7 (Factoring Xqk − X) Consider the polynomial Xqk − X ∈ Fq[X]. Thispolynomial factors in distinct linear factors in the extension Fqk [X]. (Ingeneral: the finite field with q elements consists precisely of the roots ofXq−X.) This exercise is mainly about the statement that Xqk −X ∈ Fq[X]factors over Fq[X] as the product of all monic irreducible polynomials whosedegree divides k.

(a) Let f ∈ Fq[X] be an irreducible polynomial of degree m > 0. Provethat f divides Xqm − X. [Hint: f has at least one zero in the fieldFq[X]/(f); argue that gcd(f, Xqm − X) 6= 1.]

(b) Let f ∈ Fq[X] be an irreducible divisor of Xqk − X. Show that thefield Fq[X]/(f) is contained in Fqk and conclude that the degree of f

divides k. [Hint: when does Xqm − X divide Xqk − X?]

(c) Let f ∈ Fq[X] be an irreducible polynomial whose degree divides k.

Prove that f divides Xqk −X. Conclude that Xqk −X ∈ Fq[X] factorsover Fq[X] as the product of all monic irreducible polynomials whosedegree divides k.

(d) Suppose f is a squarefree polynomial of degree m. How would youuse gcd computations to find the product of the irreducible factors ofdegree exactly k?

32 Exercises

8 (Irreducibility criteria) Let f ∈ Fq[X] be a squarefree polynomial ofdegree n > 0. Prove the following statements.

(a) If gcd(f, Xqi − X) = 1 for i = 1, 2, . . . , [n/2], then f is irreducible.[Hint: assume f is reducible and use Exercise 7.]

(b) If f is a divisor of Xqn −X and gcd(f, Xqn/` −X) = 1 for every prime` dividing n, then f is irreducible.

(c) If the matrix A from Exercise 6 has rank n − 1, then f is irreducible.

1.7 Week 7 33

1.7 Week 7

1.7.1 Resultants

1.7.1 Let f, g ∈ k[X] be two polynomials of positive degree. The resultant of f, gis an element of k that determines if f and g have a common factor or not:f and g have a nonconstant common factor if and only if the resultant is 0.If f and g have a common factor h, then there is a polynomial relation ofthe form Af + Bg = 0 with deg(A) < deg(g) and deg(B) < deg(f): simplytake A = g/h and B = −f/h. The following lemma states that the conversealso holds.

1.7.2 Lemma. Let f and g be of positive degree. Then f and g have a factor incommon if and only if there exist nontrivial polynomials A and B satisfyingdeg(A) < deg(g) and deg(B) < deg(f) such that

Af + Bg = 0.

Proof. One implication was shown above, so we assume that we have arelation Af +Bg = 0 as in the statement of the lemma and that f and g arerelatively prime. From the last assumption, we deduce from the euclideanalgorithm that there exists a relation uf + vg = 1. Multiplying this relationby B we obtain ufB+vgB = B. Using Bg = −Af we find B = ufB−vAf =(uB − vA)f contradicting the assumption deg(B) < deg(f). ¤

1.7.3 To check the existence of A and B comes down to solving a system of linearequations. If

f = f0 + f1X + · · · + fmXm (f0 6= 0, fm 6= 0) andg = g0 + g1X + · · · + gnXn (g0 6= 0, gn 6= 0),

and if we write

A = a0 + a1X + · · · + an−1Xn−1,

B = b0 + b1X + · · · + bm−1Xm−1,

then substitution in the equation Af + Bg = 0 gives a system of linear

34 Exercises

equations, whose coefficient matrix is the transpose of the (m + n)–matrix

A(f, g) =

f0 f1 · · · fm

f0 · · · fm−1 fm...f0 f1 · · · fm

g0 g1 · · · gn

g0 · · · gn...g0 · · · gn

This system of equations has a nontrivial solution if and only if the deter-minant det(A(f, g)) = 0.

Note that if the resultant is 0, the gcd of f and g is a non-trivial commonfactor.

1.7.4 Definition. The determinant of the matrix in (1.7.3) is called the resul-tant of f and g and denoted by R(f, g). The notation R(f, g, x) is used toemphasise that we view f and g as polynomials in x.

1.7.5 Remark. Although not stated explicitly, the resultant is also useful forpolynomials in several variables. For instance, the polynomials X3 +XY 2 +Y X2 + Y 3 and X + Y can be viewed as polynomials in x with coefficientsin Q (y): Y 3 + (Y 2) X + Y X2 + X3 and Y + X. The resultant is

∣

∣

∣

∣

∣

∣

∣

∣

Y 3 Y 2 Y 1Y 1 0 00 Y 1 00 0 Y 1

∣

∣

∣

∣

∣

∣

∣

∣

= 0,

showing that the two polynomials have a common factor in Q (Y )[X]. Byclearing denominators we obtain a common factor in Q [X, Y ].

1.7.6 Example. Let f = ax2 + bx + c with a 6= 0. Then

R(f, f ′) =

∣

∣

∣

∣

∣

∣

c b ab 2a 00 b 2a

∣

∣

∣

∣

∣

∣

= −a (b2 − 4ac).

In this expression, the well-known discriminant b2−4ac of a quadratic poly-nomial occurs. We regain the result that f has a zero of multiplicity 2 ifand only if b2 − 4ac = 0.

1.7 Week 7 35

1.7.7 Proposition. Let f, g ∈ k[X] be polynomials of positive degree.

a) f and g have a common factor if and only if R(f, g) = 0.

b) R(f, g) ∈ (f, g), i.e., there exist polynomials u and v such that uf +vg = R(f, g).

Proof. The first item was shown above, so we turn to b). If R(f, g) = 0,then we can take u = v = 0. If R(f, g) 6= 0, then f and g are relativelyprime, so there exist u and v with uf + vg = 1. Multiplying through byR(f, g) gives the required identity. ¤

1.7.8 For computations, the determinant in the definition of the resultant is quiteinefficient. A more efficient way to compute the resultant exists.

1.7.2 Exercises

1 (Using Hensel’s Lemma) Find the factorization of X3+2X2−3 ∈ Q[X]using Hensel’s lemma and starting from the factorization mod 2.

2 (Cyclotomic polynomials) Factor Xm − 1 in Q[X] for various positiveintegers m (use a computer algebra package). See any pattern? Relate thedegrees of the various factors to the Euler φ function.

3 (Bound on the resultant) Let f, g be non-zero polynomials in Z[X].Prove that |res(f, g)| ≤ ||f ||deg(g) · ||g||deg(f).

4 Suppose you want to compute the resultant res of two polynomials f andg. Let lc(f) denote the leading coefficient of f . An algorithm to computethe resultant is:

Input: f , gOutput: resh := f ; s := g; res := 1

WHILE deg(s) > 0 DOr := remainder(h, s); res := (−1)deg(r) deg(s) lc(s)deg(h)−deg(r)resh := s; s := r

IF h = 0 or s = 0 THEN res := 0 ELSEIF deg(h) > 0 THEN res := sdeg(h)res ELSE

Now, using this algorithm or not, find the resultant of the followingpolynomials:

(a) 2X2 + 3X + 1 and 7X2 + X + 3;

36 Exercises

(b) X5 − 3X4 − 2X3 + 3X2 + 7X + 6 and X4 + X2 + 1;

(c) 3X4 + 3X3 + X2 − X − 2 and X3 − 3X2 + X + 5.

5 Recall that if f and g are polynomials in K[X] then f and g have a commonfactor if and only if Res(f, g, X) = 0.

Denote by disc(f) the discriminant of f , i.e., the resultant of f and f ′.Find the discriminant of f in a) and b) below. Then show that f has amultiple root if, and only if, disc(f) = 0.

(a) f(x) = aX2 + bX + c.

(b) f(x) = a0X3 + a2X

2 + a3X + a4.

(c) How can you tell that

6X4 − 23X3 + 32X2 − 19X + 4

has a multiple root in C?

6 Let K be a differential field with derivation D, i.e., a field K of characteristic0 with a map D : K → K satisfying for all f, g ∈ K:

• D(f + g) = Df + Dg,

• D(fg) = f · Dg + g · Df .

Such a map D is called a derivation or differential operator . This definitionis intended to capture the properties of differentiation within an algebraicsetting. Prove the following basic properties:

(a) D(0) = D(1) = 0;

(b) D(−f) = −D(f) for all f ∈ K;

(c) D(αf + βg) = αD(f) + βD(g) for all f , g, α, β in K with D(α) =D(β) = 0;

(d) D(f

g) =

D(f) · g − f · D(g)

g2for all f , g in K with g 6= 0;

(e) D(fn) = nfn−1D(f) for all n ∈ Z and f ∈ K∗.

1.7 Week 7 37

7 Prove that the field of constants of the differential field Q(X) with ordinarydifferentiation is Q. [Hint: Use part (d) from the previous problem, startingfrom a quotient p/q with p and q relatively prime; then investigate D(P ) =(p · D(q))/q.]

8 Express arctan(x), arcsin(x) and arccos(x) in terms of the log function.[Hint: use a partial fraction decomposition for the derivative of the arctan;use the exponential expressions of sin and cos for the other two.] Don’tworry about domains of definition!

38 Exercises

1.8 Week 8

Here are some definitions, etc., on symbolic integration, relevant for this andnext week’s exercises.

• (Squarefree factorization) The squarefree factorization of a poly-nomial p (of positive degree) in K[X] is

p = p1 · p22 · · · · · pk

k,

where each pi is squarefree (i.e., has no repeated factors) and wheregcd(pi, pj) = 1 for i 6= j.

• (Differential field extension) Let (L, DL) and (K, DK) be differen-tial fields, where L is a field extension of K. If DL(f) = DK(f) for allf ∈ K, then (L, DL) is called a differential extension field of (K, DK).If no confusion arises, we simply say that L is a differential extensionfield of K.

• (Elementary extensions) There are three types of elementary ex-tensions (note that these three types need not be really disjoint: forinstance, θ may be both logarithmic and transcendental):

– logarithmic: θ = log(u) with θ′ = u′/u;

– exponential : θ = exp(u) with θ′/θ = u′;

– algebraic/transcendental : θ is algebraic if it satisfies a polynomialequation, θ is transcendental if it is not algebraic.

A differential field extension L of K iselementary if L is of the formL = K(θ1, . . . , θn), where for each i = 1, . . . , n, the element θi is eitherlogarithmic or exponential or algebraic over K(θ1, . . . , θi−1).

• (Hermite’s method) Let p, q ∈ K[x] be relatively prime and let qbe monic. Hermite’s method consists of reducing the integral of p/qto an expression of the form

r

s+

∫

a

b,

where a, b, r, s ∈ K[x], deg(a) < deg(b), and b is monic and squarefree.

• (Rothstein–Trager) Let K(x) be a differential field in one variablewith field of constants K. Let a, b ∈ K[x] be relatively prime with

1.8 Week 8 39

deg(a) < deg(b) and with b monic. Then the minimal algebraic exten-sion of K such that the integral of a/b can be expressed as

∫

a

b=

∑

i

ci log(vi),

with ci ∈ K and vi ∈ K[x], is the splitting field K of the resultant

R(z) = resx(a − z D(b), b) ∈ K[z].

In fact, the ci are the roots of R(z) and vi = gcd(a − ci D(b), b) forevery i.

• (Liouville’s principle) Suppose the integral of f ∈ K exists in anelementary extension L of K, which has the same field of constants asK. Then

∫

f = v0 +∑

i

ci log(vi) or f = v′0 +∑

i

civ′ivi

for some constants ci and elements vi ∈ K.

• (Criterion for being elementary: single logarithmic extension)Let K be a differential field with field of constants C and let K(θ) bea transcendental logarithmic extension of K with the same field ofconstants. Suppose p(θ)/q(θ) satisfies

a) gcd(p(θ), q(θ)) = 1;

b) deg p(θ) < deg q(θ);

c) q(θ) is monic and squarefree.

Then∫

p(θ)

q(θ)

is elementary if and only if the zeros of R(z) = Resθ(p(θ)−zq(θ)′, q(θ))are constants. In case the integrak is elementary, then

∫

p(θ)

q(θ)=

m∑

i=1

ci log(vi(θ)),

where the ci are the distinct roots of the resultant R(z) and where vi

equals gcd(p(θ)− ciq(θ)′, q(θ)). In fact, K = K(c1, . . . , cm) is the min-

imal algebraic extension of K such that the integral can be expressedin such a form.

40 Exercises

• (Criterion for being elementary: single exponential exten-sion) Let K be a differential field with field of constants C and letK(θ) be a transcendental exponential extension of K with the samefield of constants. Suppose p(θ)/q(θ) satisfies

a) gcd(p(θ), q(θ)) = 1 and θ does not divide q(θ);

b) deg p(θ) < deg q(θ);

c) q(θ) is monic and squarefree.

Then∫

p(θ)

q(θ)

is elementary if and only if the zeros of R(z) = Resθ(p(θ)−zq(θ)′, q(θ))are constants.

Again, there is an explicit form if the integral is elementary. Letθ′/θ = u′, then the explicit form is

−∑

i

ci deg(vi(θ))u +∑

i

ci log(vi),

where the ci are the distinct roots of the resultant R(z), and vi =gcd(p(θ)−ciq(θ)

′, q(θ)). The minimal finite extension of K over whichsuch an explicit form exists is K with the ci adjoined.

• Note: there are also theorems dealing with the case of an algebraicextension.

Exercises

1 In a) and b) find the squarefree factorizations of the indicated polynomialsin Q[X].

(a) X5 + 3X4 + 4X3 + 4X2 + 3X + 1.

(b) ((X − 1)X2)2.

(c) Suppose p = p1 · p22 · · · · · pk

k is the squarefree factorization of p ∈ Q[X].Suppose L is a field extension of Q and consider p ∈ L[X]. What isthe squarefree factorization of p? [Hint: irreducible polynomials haveno multiple roots, see Week 6, Exercise 5.] Compare with the usualfactorization into irreducibles.

1.8 Week 8 41

2 (Partial fraction decomposition) In this exercise we will construct analgorithm to find partial fraction decompositions using gcd computations.

a) Let b, d1, d2 be polynomials (of positive degree) such that b = d1d2,and such that gcd(d1, d2) = 1. Use the extended euclidean algorithmto express any fraction a/b in the form

a

b= a0 +

a1

d1+

a2

d2,

where the ai’s are polynomials, and, for i = 1, 2, the polynomials ai

satisfy either ai = 0 or deg(ai) < deg(di).

b) Suppose b is written as a product d1 · · · dn of relatively prime polyno-mials di. Use the previous item to produce a partial fraction decom-position of a/b of the form

a

b= a0 +

a1

d1+ · · · + an

dn,

where the ai are polynomials such that for each i = 1, . . . , n, eitherai = 0 or deg(ai) < deg(di).

(c) Outline an algorithm to produce partial fraction decompositions, basedon the previous items. Illustrate it in the case

X

(X2 + 1)(X2 + X + 1).

Determine an antiderivative of this expression.

3 (Squarefree factorizations) The derivative of the polynomial p is de-noted by D(p) or p′.

(a) Show that any monic polynomial p in K[X] of positive degree has asquarefree factorization. [Hint: factorization in irreducibles.]

(b) Suppose f = f e1

1 · · · f err is the factorization of f into irreducibles fac-

tors. Determine gcd(f, f ′) and f/ gcd(f, f ′). Show how to extract fromthese expressions the product of the irreducible divisors occurring inf with exponent exactly 1.

(c) Starting from the observations in (b) indicate how to produce thesquarefree factorization using only gcd computations and derivatives(and not factorization into irreducibles!).

42 Exercises

(d) Show that the following algorithm produces squarefree factorizations.

Input: p(X)

Output: a(X)

i := 1; a(X) := 1; b(X) := p′(X); q(X) := gcd(p(X), b(X)); w(X) :=p(X)

q(X)

WHILE q(X) 6= 1 DO

y(X) := gcd(w(X), q(X)); z(X) :=w(X)

y(X)

a(X) := a(X) · z(X)i; i := i + 1

w(X) := y(X); q(X) :=q(X)

y(X)

a(X) := a(X) · w(X)i

4 Let α ∈ C be an algebraic number, i.e., α satisfies a monic polynomialequation over Q. Suppose α belongs to a differential field extension of Q(x).Show that D(α) = 0.

5 Describe the differential field extensions of Q(x) needed to work with theindicated expressions. Indicate whether they are exponential, logarithmicor algebraic. For example, for f = ex + e2x you need Q(x, θ, η) with θ = ex,η = e2x. However, η = θ2, so Q(x, θ, η) = Q(x, θ) and this is an exponential(transcendental) extension of Q(x). For f =

√x you need Q(x,

√x), which

is an algebraic extension of Q(x) since√

x satisfies the equation Y 2 − x = 0in (Q(x))[Y ].

(a) f = exp(x) + exp(x/2).

(b) f = log(x2 + 1)log(x).

(c) f =√

log(x2 + 1)log(x). Relate this to the previous item.

6 In each of the following cases, indicate which field extension of Q is neededto express the antiderivative.

(a) f = 1x2−2

(b) f = 1x2+2

1.8 Week 8 43

7 In each of the following cases compute∫

(a/b) by first solving the equationRes(a − zb′, b) = 0 in z.

(a) a = 1, b = x3 + x.

(b) a = 1, b = x2 − 3.

8 Explain in detail how to compute the following integrals and give all inter-mediate results of your computation.

(a)∫

x7 − 2x3 + x2 − x + 6

2x5 − 14x4 + 20x3 + 20x2 − 14x + 2.

(b)∫

x5 − x4 + 4x3 + x2 − x + 5

x4 − 2x3 + 5x2 − 4x + 4.

9 Decide if the following are elementary, and if so, determine the integral.

(a)∫

1

log(x2 + 1).

(b)∫

2x

(x2 + 1) log(x2 + 1).

(b)∫

3x2 − 6x + 1

(x3 − 3x2 + x − 3) log(x3 − 3x2 + x − 3).

10 Decide if the following are elementary, and if so, determine the integral.

(a)∫

1

exp(x) + 1.

(b)∫

exp(x3) .

44 Exercises

1.9 Week 9

1 Let K be a differential field. Show that the subset of constants,

{c ∈ K | D(c) = 0},

is a subfield.

2 Determine∫

1

(x2 + 1)2

by applying the extended euclidean algorithm to x2 + 1 and its derivative,and by using integration by parts.

3 Let L be a differential field extension of K and let θ ∈ L be transcendentaland logarithmic over K.

(a) Show that it makes sense to speak of the degree in θ of a polynomialexpression in θ with coefficients in K.

(b) Let a(θ) be such a polynomial expression of positive degree. Showthat

deg(a(θ)′) =

{

deg(a(θ)) − 1 if the leading coefficient is a constantdeg(a(θ)) otherwise.

4 (Transcendence of the logarithm) In this exercise we show that thereis no rational function f ∈ Q(x) such that f ′ = 1/x (i.e., the logarithm doesnot belong to Q(x).

(a) Suppose f = p/q ∈ Q(x) satisfies D(f) = 1/x. Assume p, q ∈ Q[x] andassume gcd(p, q) = 1. Show that x | q.

(b) Write q = xnq with gcd(x, q) = 1. Substitute for q and derive that x | p.Conclude the proof.

(c) Show that the logarithm log(x) is transcendental over Q(x), i.e., is not azero of a polynomial Y n + a1Y

n−1 + · · · + an ∈ Q(x)[Y ].

5 (Transcendence of the exponential) The exponential function eax, witha ∈ Q\{0}, does not belong to Q(x). Assume f = p/q, with p, q ∈ Q[x] andrelatively prime, satisfies D(f) = af .

(a) Show that this leads to a contradiction.

1.9 Week 9 45

(b) Show that ex is transcendental over Q(x), i.e., satisfies no polynomialequation of the form

Y n + a1Yn−1 + · · · + an = 0,

where a1, . . . , an ∈ Q(x).

(c) Let K : Q be an algebraic extension of Q. Is ex still transcendentalover K(x)? [Hint: what is the derivative of an algebraic number?]

(d) Show that exp(x2) is transcendental over Q(x). [Hint: use the inter-mediate fields Q(x2) and Q(x2, exp(x2)).]

6 Let L be a differential field extension of K. The element θ ∈ L is expo-nential over K if θ′/θ = u′ for some u ∈ K. We write θ = exp(u) in thatcase.

(a) In fact, we should say: θ is an exponential of u. Comment on thisremark using the expressions u + c, with c′ = 0 and 3θ.

(b) Given the ambiguity as remarked upon in (a), show that exp(u+ v) =exp(u) · exp(v).

(c) Let u ∈ K. Show that {θ ∈ L | θ′ = θu′} is a linear subspace of L overthe constants.

Index

algebraic extension, 35algebraic number, 16

Circle theorem of Apollonius, 18

derivation, 33differential extension field, 35differential field, 33differential operator, 33

Eisenstein’s criterion, 24elementary extension, 35elementary symmetric polynomial,

20exponential extension, 35

formal derivative, 27

Grobner basisminimal, 13

intersection of ideals, 19

Liouville’s principle, 36logarithmic extension, 35

minimal polynomial, 16monomial ordering, 4

Nullstellensatz, 15

partial order, 4

radical, 14radical membership problem, 15

resultant, 31Rothstein–Trager theorem, 35

squarefree, 27sum of ideals, 19symmetric polynomial, 20

total order, 4transcendental extension, 35

well–ordering, 4

46