SAN JOSÉ STATE UNIVERSITY
Thayer Watkins

Galois Theory Without Polynomials

Galois theory is associated with the solutions to polynomial equations, but the fundamental theory is about fields and subfields and its application to the solutions of polynomial equations is merely a special instance of the theory. This document is an attempt to explain and illustrate Galois theory. While the proof is not dependent upon polynomial theory there is no reason to exclude polynomials from the examples and illustrations.

The highlighted words are hyperlinks to explanations and examples of the concepts involved. This work is still in the process of construction.

Fundamental Theorem of Galois Theory

Let K be a field and F be a subfield of K. If there exits an automorphism group G of K such that:

##### F = KG
then:
1. There is a one-to-one correspondence between the subfields of K which contain F and and the subgroups of the group G of automorphisms.
2. Furthermore,
##### [K:F] = |G|, i.e., the rank of K over F equals the cardinality of the automorphism group G and G = Gal(K/F), i.e., the automorphism group is isomorphic to the Galois group of K over F.
3. The one-to-one correspondence associates to any subfield L the Galois group Gal(K/L)
4. The one-to-one correspondence associates to any subgroup H of G its field of invariants.

# Concepts and Definitions

### Fields

A field is a set with two operations, say [S,+,*] such at [S,+] is an abelian group and (*) is associative, and (*) is left and right distributive over (+). The exists an identity element for (*). For every element of S except the identity of (+) there exists an inverse with respect to (*).

#### Subfield

A set [T,+,*] is a subfield of the field [S,+,*] if T is a subset of S and [T,+,*] is a field.

### Morphism

A morphism is a mapping from one algebraic structure to another that preserves the relationships of the elements with respect to the operations of the structures. If [K,+,*] and [L,&,^] are fields, then a mapping f:K→L is a homomorphism if and only if (iff)
##### f(a+b)=f(a)&f(b) and f(a*b)=f(a)^f(b)
then f is a homomorphism of the fields. If the mapping is of a set S onto itself then it is called an isomorphism.

## Notation

• Rank: The rank [K:F] is the dimension of K considered as a vector space over F.
• Galois Group: Gal(K/F) is the group of field-automorphisms of K which leave the elements of F invariant. This is referred to as the Galois group of K over F.