📂Abstract Algebra

Galois Theory

Theorem¹

Let $K$ be a finite normal extension field of $F$, and let $F \le E \le K$. Denote by $\lambda (E)$ the subgroup of $G ( K / F )$ that leaves $E$ fixed. Then the map $\lambda$ is an isomorphism mapping every $E$ between $F$ and $K$ to every subgroup of $G ( K / F )$. $\lambda$ has the following properties.

1. $\lambda ( E ) = G ( K / E )$
2. $E = K_{ G ( K / E ) } = K_{ \lambda (E) }$
3. For $H \le G ( K / F )$, $\lambda ( K_{H} ) = H$
4. $[K : E] = | \lambda (E) |$ and $[ E : F ] = \left( G ( K / F ) : \lambda (E) \right)$
5. $E$ is a normal extension field of $F$ $\iff$ $\lambda (E)$ is a normal subgroup of $G (K / F)$.
6. If $\lambda (E)$ is a normal subgroup of $G ( K / F )$, then $G (E / F) \simeq G ( K / F ) / G ( K / E )$

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• $[ E : F ]$ means the degree.
• $G(E / F)$ means the group of $E$ over $F$.
• $\left( G ( K / F ) : \lambda (E) \right)$ means the index in group theory.
• $K_{ \lambda (E) }$ is the set consisting only of the elements of $K$ fixed by $\lambda (E)$.