§ Galois extension

  1. MM is normal and separable (over KK).
  2. deg(M/K)=Gdeg(M/K) = |G|. We will show that Gdeg(M/K)|G| \leq deg(M/K). So MM is "symmetric as possible" --- have the largest possible galois group
  3. K=MGK = M^G [The fixed poits of MM under GG]. This is useful for examples.
  4. MM is the splitting field of a separable polynomial over KK. Recall that a polynomial is separable over KK if it has distinct roots in the algebraic closure of KK. Thus, the number of roots is equal to the degree.
  5. KLMK \subseteq L \subseteq M and 1HG1 \subseteq H \subseteq G: There is a 1-1 correspondece between LGal(M/L)L \mapsto Gal(M/L) [NOT L/KL/K! ], and the other way round, to go from HH to MHM^H. This is a 1-1 correspondence. LL is in the denominator because we want to fix LL when we go back.

§ (4) implies (1)