Galois Theory: A Study of Cyclotomic Field ... - Scripps College

Introduction 9
is, L = L ′ = K. Thus, σ is an automorphism of K. Since σ is the identity
map on F , b stays fixed (it is equal to −(α + α ′ )). Thus σ must send α to
α ′ . We can conclude that σ 2 is the identity map on K since σ(α) = α ′ and
σ(α ′ ) = α.
Definition. Let K be an extension field. An automorphism of K which
is the identity on F is an F-automorphism. It has the property that σ(a) = a
for all a ∈ F . The F -automorphisms of K form a group under composition.
The group of all F -automorphisms of the field extension K is known as the
Galois group of K over F , denoted G(K/F ).
Theorem 3. For any algebraic field extension K/F, the order of the Galois
group, |G(K/F )|, divides the degree of the extension [K : F ].
Proof. See [A, p.540]
We know for any σ ∈ G(K/F ), any a in F will be fixed by σ, that is,
σ(a) = a for all a in F . But a priori, there may be other elements of K fixed
by σ.
Definition. If the order of the Galois group is equal to the degree, then
the field extension K/F is a Galois extension.
Definition. The set of elements in a field K that are fixed by all the
elements of G, a group of automorphisms of K is known as a fixed field of
G and is denoted K G .
Corollary 4. Let K/F be a Galois extension. If its Galois group is G =
G(K/F ), then F is the fixed field of G.
Proof. Let L be the fixed field, K G , of G. Clearly, F ⊂ L. Since F ⊂ L,
every L-automorphism will also be an F -automorphism. This implies that
G(K/L) is contained in G. However, every element in G is by definition of