📂Abstract Algebra

Separable Extension Field

Definition¹

Let $E$ be an extension field of $F$.

1. The number of isomorphisms of $E$ onto a subfield of $\overline{F}$ that leave $F$ fixed is called the index of $E$ over $F$, denoted by $\left\{ E : F \right\}$.
2. When $E$ is a finite field, if $\left\{ E : F \right\} = [ E : F ]$, then $E$ is called a separable extension field of $F$.
3. If $f ( \alpha )$ is a separable extension field of $F$, then $\alpha \in \overline{F}$ is said to be separable over $F$.
4. If every zero of $f(x)$ is separable over $F$, then the irreducible element $f(x) \in F [ x ]$ is said to be separable over $F$.
5. When $K$ is a finite extension of $F$, if $K$ is a separable splitting field over $F$, then $K$ is called a finite normal extension field of $F$.

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• $[ E : F ]$ denotes the degree.
• $G(E / F)$ denotes the group of $E$ over $F$.

Explanation

As an example of the index, consider $\mathbb{Q} ( \sqrt{2} , \sqrt{3} )$: the automorphisms $$ I, \psi_{\sqrt{2} , -\sqrt{2}}, \psi_{\sqrt{3} , -\sqrt{3}}, \left( \psi_{\sqrt{2} , -\sqrt{2}} \psi_{\sqrt{3} , -\sqrt{3}} \right) $$ leave $\mathbb{Q}$ fixed, so $\left\{ \mathbb{Q} ( \sqrt{2} , \sqrt{3} ) : \mathbb{Q} \right\} = 4$.

The reason separable extension fields are defined separately is that in general $\left\{ E : F \right\} \mid [ E : F ]$ holds, but there is no guarantee that the two are always equal.