4H Galois Theory 2003-4: revision guide

Below is a guide to the major topics deemed examinable. Where a result is specified it is intended that statement and proof are deemed examinable, whereas the phrase 'statement' implies that the proof is not deemed examinable. Many important ideas are discussed in the problem sets and some of these may come up in the exam. Feel free to ask me for more information.

AJB (15/03/2004)

Chapter 1

Basic properties of integral domains and fields, characteristics, Idiot's Binomial Theorem, fields of fractions, polynomial rings, homomorphism extension property, ideals and quotient rings for polynomial rings in one variable over a field.

Statements of the Eisenstein test for irreducibility of polynomials over Z, Q and k[T], k(T) with k a field, cyclotomic polynomials over Q.

Chapter 2

Basic properties of fields and their extensions, multiplicativity of dimensions of iterated extensions, simple and finitely generated extensions, examples of extensions inside C.

Chapter 3

Algebraic extensions, minimal polynomials, primitive elements, definition of an algebraic closure of a field, Kronecker's Theorem (both versions), splitting fields, quadratic polynomials and quadratic extension fields.

Monomorphisms between extension fields, effect of a monomorphism on roots of a polynomial, relationship between roots of an irreducible polynomial and monomorphisms into algebraically closed fields.

Statements of existence of algebraic closures, statement of the Extension Theorem, conjugates and roots of the minimal polynomial of an algebraic element.

Multiplicity of roots of a polynomial and separability, the derivative test, separability in characteristic 0.

The Primitive Element Theorem. Determining the minimal polynomial of a primitive element in a simple extension.

Normal extensions and splitting fields.

Chapter 4

Galois extensions, action of a Galois group on roots of polynomials and realisation as groups of permutations especially for irreducible polynomials, calculation of Galois groups over subfields of C of small degrees.

Statement of the Galois Correspondence and the main Theorem of Galois Theory.

Use of the discriminant to determine whether a Galois group of an irreducible polynomial lies in the alternating subgroup of the symmetric group.

Statement and use of Kaplansky's Theorem for quartic polynomials over Q.

Galois extensions in C and complex conjugation.

Chapter 5

Galois extensions in positive characteristic, existence and uniqueness for finite subfields of an algebraically closed field, Galois groups of finite extensions of finite fields, definition of trace and norm mappings, Galois group of the splitting field of an irreducible polynomial of form X^p-X-a over a finite field.

Chapter 6

Statement of the Fundamental Theorem of Algebra for C and the algebraic closure of Q in C.

Statement's of Artin's Theorem on linear independence of characters and Hilbert's Theorem 90.

Cyclotomic extensions over Q, their Galois groups and real subextensions, radical and Kummer extensions.