## MAS438 Fields

Semester 1, 2016/17 | 10 Credits | ||||

Lecturer: | Prof Vladimir Bavula | Reading List | |||

Aims | Outcomes | Teaching Methods | Assessment | Full Syllabus |

A field is a set where the familiar operations of arithmetic are possible. It often happens, particularly in the theory of equations, that one needs to extend a field by forming a bigger one. The aim of this course is to study the idea of field extension and various problems where it arises. In particular, it is used to answer some classical problems of Greek geometry, asking whether certain geometrical constructions, such as angle trisection or squaring the circle, are possible. The finite fields and their groups of automorphisms are studied.

**Prerequisites:** MAS220 (Algebra)

**Not with:** MAS333 (Fields)

No other modules have this module as a prerequisite.

## Outline syllabus

- Field extensions
- Factorization of polynomials
- Simple field extensions
- Towers of fields
- Ruler and compass constructions
- Finite fields and their groups of automorphisms
- The algebraic closure of a finite field
- The Frobenius automorphism

## Aims

- To illustrate how questions concerning the complex roots of real or rational polynomial equations can quickly lead to the study of subfields of the field of complex numbers
- To consolidate previous knowledge of field theory and vector space theory
- To illustrate how the general mathematical theory of vector spaces can be used to good effect in the theory of field extensions
- To illustrate how the theory of dimensions of vector spaces can be used to prove that certain ruler and compass constructions are impossible
- To illustrate the relevance of factorization of polynomials to the theory of algebraic field extensions
- To study the finite fields and their groups of automorphisms

## Learning outcomes

- use the subfield criterion to determine whether a given subset of a field is a subfield;
- understand the distinction between algebraic and transcendental elements of a field extension of a base field;
- use Eisenstein's irreducibility criterion for polynomials;
- use the minimal polynomial of an algebraic element α of a field extension of a base field F to express the inverse of a non-zero element of F(α) in the from p(α) for some polynomial p with coefficients in F;
- work with primitive n-th roots of unity and the cyclotomic
polynomial
φ
_{n}; - prove that φ
_{p}is irreducible when p is a prime number; - understand what is meant by a constructible point of the Euclidean plane and a constructible real number;
- know and be able to use a field-theoretic criterion for determining whether a given point of the Euclidean plane is constructible;
- use the theory of degrees of field extensions to prove that certain ruler and compass constructions are impossible;
- know for which integers n ≥ 3 the regular n-gon can be constructed;
- understand the concept of the group of automorphisms of a field extension;
- to be able to describe all the finite fields and their groups of automorphisms;
- understand the concept of the algebraic closure of a finite field.

## Teaching methods

Lectures, problem solving

20 lectures, no tutorials

## Assessment

One formal 2.5 hour written examination. Format: 4 questions from 4.

## Full syllabus

**1. Field extensions**

(4 lectures)

**Z**

_{p}when p is a prime number; finite fields; field of rational functions. Subfields, extension fields; subfield criterion. Every subfield of

**R**or

**C**contains

**Q**. The subfield of a field L generated by a subset X of L. Field adjunction; examples; simple field extensions.

**2. Factorization of polynomials**

(3 lectures) Reducible polynomials; irreducible polynomials; primitive polynomials in

**Z**[x]; Gauss's Lemma; the important corollary that a primitive polynomial in

**Z**[x] which is irreducible in

**Z**[x] is irreducible in

**Q**[x]. A monic cubic f(x) ∈

**Z**[x] with no integer root is irreducible over

**Q**. Eisenstein's irreducibility criterion; examples; use of the criterion in conjunction with the fact that, for a non-constant f ∈

**Z**[x] and k ∈

**Z**, f(x) is irreducible if and only if f(x+k) is; use of Eisenstein's criterion in conjunction with the fact that f(x) is irreducible if and only if x

^{n}f(1/x) is, where n is the degree of f. Primitive n-th roots of unity, where n ∈

**N**; the cyclotomic polynomial φ

_{n}; φ

_{p}is irreducible if p is prime.

**3. Simple field extensions**

(3 lectures) Elements of a field extension which are algebraic or transcendental over the base field K; simple algebraic extensions; simple transcendental extensions; minimal polynomial of an algebraic element a, and its irreducibility; if f(a) = 0 for a monic irreducible polynomial f(x) ∈ K[x], then f(x) is the minimal polynomial of a over K; use of the minimal polynomial to express the inverse of a non-zero element of K(a) as g(a) for some polynomial g(x) ∈ K[x].

**4. Towers of fields**

(3 lectures) The degree of a field extension; finite extensions; the dimension formula [M:K] = [M:L][L:K] when L is an intermediate field between K and M. The degree of a simple field extension; examples. The set of algebraic numbers is a subfield of

**C**. Splitting fields (of polynomials over subfields of

**C**).

**5. Ruler and compass constructions**

(7 lectures) Constructible points of

**R**

^{2}; constructible real numbers; examples. The classical problems: doubling the cube; trisecting angles; squaring the circle; constructing regular polygons. Standard constructions. The field of constructible real numbers. Quadratic extensions. Necessary and sufficient (field theoretic) conditions for (a,b) ∈

**R**

^{2}to be a constructible point. If a ∈

**R**is constructible, then a is algebraic over

**Q**and [

**Q**(a):

**Q**] is a power of 2. Solutions of the classical problems: statement (no proof, but some history) that π is not algebraic over

**Q**. Constructible complex numbers. Fermat primes. If p is an odd prime which is not a Fermat prime, then the regular p-gon cannot be constructed. Statement of converse, and proof for p = 5, 17. Determination of those integers n ;eq 3 for which the regular n-gon can be constructed.

**6. Finite fields and their groups of automorphisms**

Groups of automorphisms of field extensions. Finite fields and their groups of automorphisms. The Frobenius automorphism. The algebraic closure of a finite field. A classification of finite fields. The fixed field of an automorphism.

## Reading list

Type | Author(s) | Title | Library | Blackwells | Amazon |
---|---|---|---|---|---|

B |
Allenby | Rings, fields and groups | 512.8 (A) | Blackwells | Amazon |

B |
Fraleigh | A first course in abstract algebra | 512.8 (F) | Blackwells | Amazon |

B |
Herstein | Abstract algebra | 512.8 (H) | Blackwells | Amazon |

B |
Stewart | Galois theory | 512.43 (S) | Blackwells | Amazon |

(

**A**= essential,

**B**= recommended,

**C**= background.)

Most books on reading lists should also be available from the Blackwells shop on Mappin Street.