Galois Theory
Theorem 1
Let $K$ be a Galois extension of $F$ with group $F \le E \le K$. Let us denote by $\lambda (E)$ the subgroup of $G ( K / F )$ that fixes $E$. Then, the map $\lambda$ becomes an isomorphism mapping all $E$ between $F$ and $K$ to all subgroups of $G ( K / F )$. $\lambda$ has the following properties:
- $\lambda ( E ) = G ( K / E )$
- $E = K_{ G ( K / E ) } = K_{ \lambda (E) }$
- For $H \le G ( K / F )$, $\lambda ( K_{H} ) = H$
- If $[K : E] = | \lambda (E) |$ and $[ E : F ] = \left( G ( K / F ) : \lambda (E) \right)$
- $E$ is a normal extension of $F$, $\lambda (E)$ is a normal subgroup of $G (K / F)$.
- If $\lambda (E)$ is a normal subgroup of $G ( K / F )$, then $G (E / F) \simeq G ( K / F ) / G ( K / E )$
- $[ E : F ]$ means degree.
- $G(E / F)$ refers to the group of $E$ over $F$.
- $\left( G ( K / F ) : \lambda (E) \right)$ means index in group theory.
- $K_{ \lambda (E) }$ is the set of elements that are fixed from $K$ to $\lambda (E)$.
Fraleigh. (2003). A first course in abstract algebra(7th Edition): p451. ↩︎