Abstract Algebra-The Basic Graduate Year (6).pdf

page 1 of Chapter 6
CHAPTER 6 GALOIS THEORY
Fixed Fields and Galois Groups
Galois theory is based on a remarkable correspondence between subgroups of the Galois group of an
extension E/F and intermediate
elds between E and F . In this section we will set up the machinery for
the fundamental theorem. [A remark on notation: Throughout the chapter, position
of two
automorphisms will be written as a product
.]
De
nitions ments Let G =Gal(E/F ) be the Galois group of the extension E/F . If H is
a subgroup of G, the
xed
eld of H is the set of elements
xed by every automorphism in H, that is,
(H) = x E :
(x) = x for every
H .
F { ∈∈}
If K is an intermediate
eld, that is, F K E, de
ne

(K) = Gal(E/K) =
G :
(x) = x for every x K .
G { ∈∈}
I like the term “
xing group of K” for (K), since (K) is the group of automorphisms of E that leave
K
xed. Galois theory is about the relationG betweenG
xed
elds and
xing groups. In particular, the next
result suggests that the smallest sub
eld F corresponds to the largest subgroup G.
Proposition Let E/F be a
nite Galois extension with Galois group G =Gal(E/F ). Then
(i) The
xed
eld of G is F ;
(ii) If H is a proper subgroup of G, then the
xed
eld of H properly contains F .
Proof.
(i) Let F0 be the
xed
eld of G. If
is an F -automorphism of E, then by de
nition of F0,
xes
everything in F0. Thus the F -automorphisms of G coincide with the F0-automorphisms of G. Now by
() and (), E/F0 is Galois. By (), the size of the Galois group of a
nite Galois extension is the
degree of the extension. Thus [E : F ] = [E : F0], so by (), F = F0.
(ii) Suppose that F = (H). By the theorem of the primitive element (), we have E = F (
) for some
E. De
ne a polynomialF f(X) E[X] by
∈∈
f(X) = (X
(
)).


H
Y∈
If
is any automorphism in H, then we may apply
to f (that is, to the coe
cients of f; we discussed this
idea in the proof of (3