내가 헷갈려서 쓰는 대수 정의들

1. $E$ 와 $F$ 사이에 존재하는 다양한 괄호들...

$ G(E/F) :$ The group of automorphisms of $E$ leaving $F$ fixed.

$ \{ E:F \} :$ The number of isomorphisms of $E$ onto a subfield of $\bar{F}$ leaving $F$ fixed.

$ [E:F] :$ Finite extension of degree over F.

 

Finite extension + splitting $\Rightarrow$ $|G(E/F)| = \{ E:F \}$.

Finite extension + separable $\Rightarrow$ $[E:F] = \{ E:F \}$.

Finite normal$($splitting + separable$)$ extension  $\Rightarrow$ $|G(E/F)| = [E:F] = \{ E:F \}$.