Paper 2, Section II, H

Galois Theory
Part II, 2016

(a) Let KLK \subseteq L be a finite separable field extension. Show that there exist only finitely many intermediate fields KFLK \subseteq F \subseteq L.

(b) Define what is meant by a normal extension. Is QQ(1+7)\mathbb{Q} \subseteq \mathbb{Q}(\sqrt{1+\sqrt{7}}) a normal extension? Justify your answer.

(c) Prove Artin's lemma, which states: if KLK \subseteq L is a field extension, HH is a finite subgroup of AutK(L)\operatorname{Aut}_{K}(L), and F:=LHF:=L^{H} is the fixed field of HH, then FLF \subseteq L is a Galois extension with Gal(L/F)=H\operatorname{Gal}(L / F)=H.