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 = K^G

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

• 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.