Splitting Fields

(section 50) Splitting Fields

(Def 50.1) Let F be a field with algebraic closure \bar{F} . Let {f_{i} (x) | i \in I } be a collection of polynomials in F[x] . A field E \le \bar{F} is the splitting field of {f_{i} (X) |i \in I} over F if E is the smallest subfield of \bar{F} containing F and all the zeros in \bar{F} of each of the f_{i} (x) for i \in I . A field K \le \bar{F} is a splitting field over F if it is the splitting field of some set of polynomials in F[x].

(Thm 50.3) A field E, where F \le E \le \bar{F}, is a splitting field over F iff every automorphism of \bar{F} leaving F fixed maps E onto itself and thus induces an automorphism of E leaving F fixed.

(Def 50.4) Let E be an extension field of a field F. A polynomial f(x) \in F[x] splits in E if it factors into a product of linear factors in E[x].

(Cor 50.6) If E \le \bar{F} is a splitting field over F ,then every irreducible polynomial in F[x] having a zero in E splits in E.

(Cor 50.7) If E \le \bar{F} is a splitting field over F, then every isomorphic mapping of E onto a subfield of \bar{F} and leaving F fixed is actually an automorphism of E. In particular, if E is a splitting field of finite degree over F, then {E : F} = |G(E/F)|.