Page 284 - DMTH403_ABSTRACT_ALGEBRA

Unit 30: Invariant Subfield

(a) G-invariant subfield (b) variant subfield Notes
(c) finite field (d) domain subfield

2. G-invariant subfield is denoted by ...............
(a) G F (b) F G
(c) F.G (d) GF -1
3. G be a finite group of automorphisms of the field F and let K = FG then [F : K] ...............|G|.

(a)  (b) 
(c) = (d) 
4. For any subgroup H of G, we have ............... and [F : K] = [G : H]
H
(a) [F : F ] = |H| (b) [H : F]  |H|
H
F
(c) [F : H] = |H| (d) [F : H ]  |H|
–1
–1
–1
5. Let F be an algebraic extensions of the field K. Then F is said to be a ............... of K. If every
irreducible polynomial in K[x] that contains a root in F is a product of linear factors in F[x]
(a) normal extension (b) finite extension
(c) infinite extension (d) subgroup extension

30.2 Summary

The following conditions are equivalent for an extension field F of K:

(1) F is the splitting field over K of a separable polynomial;

(2) K = F for some finite group G of automorphisms of F;
G
(3) F is a finite, normal, separable extension of K.
If F is an extension field of K such that K = F for some finite group G of automorphisms
G

of F, then G = Gal(F/K).
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) If H is a subgroup of G, then the corresponding subfield is F , and
H
H = Gal(F/F ).
H
(ii) If E is a subfield of F that contains K, then the corresponding subgroup of G is
H = Gal(F/E), and
E = F .
H
(b) For any subgroup H of G, we have

[F : F ] = | H| and [F : K] = [G : H].
H
H
(c) Under the above correspondence, the subgroup H is normal if and only if the subfield
E = F is a normal extension of K. In this case,
H
Gal(E/K)  Gal(F/K) / Gal(F/E).

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