Page 273 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes Let F be a field, and let G be a subgroup of Aut(F). Then
{a  F | (a) = a for all   G}
is called the G-fixed subfield of F, or the G-invariant subfield of F, and is denoted by F .
G
(Proposition shows that F is actually a subfield of F.) If F is the splitting field over K of a
G
separable polynomial and G = Gal(F/K), then Proposition shows that F = K. Artin’s lemma
G
provides the first really significant result of the section. It states that if G is a finite group of
automorphism of the field F, and K = F , then [F : K]  |G|.
G
Let F be an algebraic extension of the field K. Then F is said to be a normal extension of K if every
irreducible polynomial in K[x] that contains a root in F is a product of linear factors in F[x]. With
this definition, the following theorem and its corollary can be proved from previous results.

Theorem 4: The following are equivalent for an extension field F of K:
(1) F is the splitting field over K of a separable polynomial;
(2) K = FG for some finite group G of automorphism of F;
(3) F is a finite, normal, separable extension of K.
As a corollary, we obtain the fact that if F is an extension field of K such that K = F for some finite
G
group G of automorphisms of F, then G = Gal(F/K).
The next theorem is the centerpiece of Galois theory. In the context of the fundamental theorem,
we say that two intermediate subfields E and E are conjugate if there exists   Gal(F/K) such
2
1
that (E ) = E . Proposition states that if F is the splitting field of a separable polynomial over K,
1
2
and K  E  F, with H = Gal(F/E), then Gal(F/(E)) = H , for any   Gal(F/K).
-1
Theorem 5 (The Fundamental Theorem of Galois Theory): Let F be the splitting field of a
separable polynomial over the field K, and let G = Gal(F/K).
(a) There is a one-to-one order-reversing correspondence between subgroups of G and subfields
of F that contain K:
(i) The subfield F corresponds to the subgroup H, and H = Gal(F/F ).
H
H
(ii) If K  E  F, then the corresponding subgroup is Gal(F/E), and E = F (F/E).
Gal
(b) [F : FH] = |H| and [F : K] = [G : H], for any subgroup H of G.
H
(c) Under the above correspondence, the subgroup H is normal iff F is a normal extension
H
of K. In this case, Gal(E/K)  Gal(F/K) / Gal(F/E).
For example, suppose that F is a finite field of characteristic p, and has p elements. Then
m
[F : GF(p)] = m, and so G = Gal(F= GF(p)) is a cyclic group of degree m by Corollary. Since G is
cyclic, the subgroups of G are in one-to-one correspondence with the positive divisors of m.
Proposition shows that the subfields of F are also in one-to-one correspondence with
the positive divisors of m. Remember that the smaller the subfield, the more automorphisms
will leave it invariant. By the Fundamental Theorem of Galois Theory, a subfield E with
[E : GF(p)] = k corresponds to the cyclic subgroup with index k, not to the cyclic subgroup of
order k.

Self Assessment

1. If F has characteristics zero, then its prime subfield is isomorphic to Q and if F has
characteristics P, for some prime number P, then its prime subfield is ................ to Zp.
(a) homomorphic (b) isomorphic
(c) automorphism (d) polynomial

266 LOVELY PROFESSIONAL UNIVERSITY