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 = K^{G}
then:
- 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.
- 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.
- The one-to-one correspondence associates to any subfield
L the Galois group Gal(K/L)
- 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.