Galois Theory

(Section 53) Galois Theory

(Recall)

Let F \le E \le \bar{F}, \alpha \in E , and let \beta be a conjugate of \alpha over F, irr(\alpha F) has \beta as a zero also. Then there is an isomorphism \psi_{\alpha , \beta } mapping F(\alpha ) onto F(\beta ) that leaves F fixed and maps \alpha onto \beta.

If F \le E \le \bar{F} and \alpha \in E, then an automorphism \sigma of \bar{F} that leaves F fixed must map \alpha onto some conjugate of \alpha over F.

If F \le E, the collection of all automotphisms of E leaving F fixed forms a group G(E/F) . For any subset S of G(E/F) , the set of all elements of E left fixed by all elements of S is a field E_{s} . Also, F \le E_{G(E/F)} .

A field E , F \le E \le \bar{F}, is a splitting field over F iff every isomorphism of E onto a subfield of \bar{F} leaving F fixed is an automorphism of E. If E is a finite extension and a splitting field over F, then |G(E/F)| = {E : F} .

If E is a finite extension of F, then {E : F} divides [E : F] . If E is also separable over F, then {E : F} = [E : F]. Also, E is separable over F iff irr( \alpha , F ) has all zeros of multiplicity 1 for every \alpha \in E.

If E is a finite extension of F and is separable splitting field over F, then |G(E/F)| = {E:F} = [E : F].

(subsection) Normal Extensions

(Def 53.1) A finite extension K of F is a finite normal extension of F if K is a separable splitting field over F.

(Thm 53.2) Let K be a finite normal extension of F, and let E be an extension of F, where F \le E \le K \le \bar{F}. Then K is a finite normal extension of E< and G(K/E) is precisely the subgroup of G(K/F) consisting of all those automorphisms that leave E fixed.

Moreover, two automorphisms \sigma and \tau in G(K/F) induce the same isomorphisms of E onto a subfield of \bar{F} iff they are in the same left coset of G(K/E) in G(K/F).

(subsection) Main Theorem

(Def 53.5) If K is a finite normal extension of a field F, then G(K/F) is the Galois group of K over F.

(Main theorem of Galois Theory) (Thm 53.6) Let K be a finite normal extension of a field F, with Galois group G(K/F) . For a field E< where F \le E \le K, let \lambda(E) be the subgroup of G(K/F) leaving E fixed. Then \lambda is a one-to-one map of the set of all such immediate fields E onto the set of all subgroups of G(K/F). The following properties hold for \lambda.

\lambda (E) = G(K/E)

E = K_{G(K/E)} = K_{\lambda (E)}

For H \le G(K/F) , \lambda(E_{H}) = H

[K : E] = |\lambda (E)} and [E : F] = (G(K/F) : \lambda(E) ), the number of left cosets of \lambda(E) in G(K/F).

E is a normal extension of F iff \lambda(E) is a normal subgroup of G(K/F). When \lambda(E) is a normal subgroup of G(K/F), then G(E/F) \simeq G(K/F) / G(K/E).

The diagram of subgroups of G(K/F) is the inverted diagram of intermediate field of K over F.

(prop) The Galois group G(K/F) is the group of polynomial f(x) over F.

(subsection) Galois Groups over Finite fields

(Thm 53.6) Let K be a finite extension of degree n of a finite field F of p^{r} elements. Then G(K/F) is cyclic of order n, and is generated by \sigam_{p^{r}} , where for \alpha \in K, \sigma_{p^{r}} (\alpha) = \alpha^{p^{r}}.