Ordered Field

  • View
    215

  • Download
    1

Embed Size (px)

DESCRIPTION

1. From Wikipedia, the free encyclopedia2. Lexicographical order

Text of Ordered Field

  • Ordered eldFrom Wikipedia, the free encyclopedia

  • Contents

    1 Finite eld 11.1 Denitions, rst examples, and basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Existence and uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Explicit construction of nite elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

    1.3.1 Non-prime elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3.2 Field with four elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.3 GF(p2) for an odd prime p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3.4 GF(8) and GF(27) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3.5 GF(16) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

    1.4 Multiplicative structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4.1 Discrete logarithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4.2 Roots of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

    1.5 Frobenius automorphism and Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6 Polynomial factorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

    1.6.1 Irreducible polynomials of a given degree . . . . . . . . . . . . . . . . . . . . . . . . . . 71.6.2 Number of monic irreducible polynomials of a given degree over a nite eld . . . . . . . . 8

    1.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.8 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

    1.8.1 Algebraic closure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81.8.2 Wedderburns little theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

    1.9 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.11 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.12 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

    2 Glossary of eld theory 112.1 Denition of a eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Basic denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.4 Types of elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Field extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6 Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

    i

  • ii CONTENTS

    2.7 Extensions of Galois theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

    3 Ordered eld 173.1 Denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

    3.1.1 Total order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1.2 Positive cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.1.3 Equivalence of the two denitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.1.4 Fan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

    3.2 Properties of ordered elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.2.1 Vector spaces over an ordered eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

    3.3 Examples of ordered elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.4 Which elds can be ordered? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.5 Topology induced by the order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.6 Harrison topology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.7 Superordered elds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

    4 Perfect eld 224.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.2 Field extension over a perfect eld . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.3 Perfect closure and perfection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.4 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.6 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234.7 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

    5 Separable extension 255.1 Informal discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2 Separable and inseparable polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265.4 Separable extensions within algebraic extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . 275.5 The denition of separable non-algebraic extension elds . . . . . . . . . . . . . . . . . . . . . . 275.6 Dierential criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295.11 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 30

    5.11.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

  • CONTENTS iii

    5.11.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.11.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

  • Chapter 1

    Finite eld

    In mathematics, a nite eld or Galois eld (so-named in honor of variste Galois) is a eld that contains a nitenumber of elements. As with any eld, a nite eld is a set on which the operations of multiplication, addition,subtraction and division are dened and satisfy certain basic rules. The most common examples of nite elds aregiven by the integers mod n when n is a prime number.The number of elements of a nite eld is called its order. A nite eld of order q exists if and only if the order q isa prime power pk (where p is a prime number and k is a positive integer). All elds of a given order are isomorphic.In a eld of order pk, adding p copies of any element always results in zero; that is, the characteristic of the eld is p.In a nite eld of order q, the polynomial Xq X has all q elements of the nite eld as roots. The non-zero elementsof a nite eld form a multiplicative group. This group is cyclic, so all non-zero elements can be expressed as powersof a single element called a primitive element of the eld (in general there will be several primitive elements for agiven eld.)A eld has, by denition, a commutative multiplication operation. A more general algebraic structure that satisesall the other axioms of a eld but isn't required to have a commutative multiplication is called a division ring (orsometimes skeweld). A nite division ring is a nite eld by Wedderburns little theorem. This result shows that theniteness condition in the denition of a nite eld can have algebraic consequences.Finite elds are fundamental in a number of areas of mathematics and computer science, including number theory,algebraic geometry, Galois theory, nite geometry, cryptography and coding theory.

    Commutative rings integral domains integrally closed domains unique factorization do-mains principal ideal domains Euclidean domains elds nite elds

    1.1 Denitions, rst examples, and basic propertiesA nite eld is a nite set on which the four operations multiplication, addition, subtraction and division (excludingby zero) are dened, satisfying the rules of arithmetic known as the eld axioms. The simplest examples of niteelds are the prime elds: for each prime number p, the eld GF(p) (also denoted Z/pZ, Fp , or Fp) of order (that is,size) p is easily constructed as