B4.3
Part II, 2001
Define the concept of separability and normality for algebraic field extensions. Suppose is a simple algebraic extension of , and that denotes the group of -automorphisms of . Prove that , with equality if and only if is normal and separable.
[You may assume that the splitting field of a separable polynomial is normal and separable over .]
Suppose now that is a finite group of automorphisms of a field , and is the fixed subfield. Prove:
(i) is separable.
(ii) and .
(iii) is normal.
[The Primitive Element Theorem for finite separable extensions may be used without proof.]