📂Abstract Algebra

Scalable Divisible Body

Definition ¹

Let’s say $E$ is an extension field of $F$.

1. The number of automorphisms from $E$ to a subfield $\overline{F}$, leaving a fixed $F$, is called the index of $E$ over $F$, denoted as $\left\{ E : F \right\}$.
2. If $E$ is a finite field and $\left\{ E : F \right\} = [ E : F ]$, $E$ 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 separable over $F$.
4. If every zero of $f(x)$ is separable over $F$, the irreducible element $f(x) \in F [ x ]$ is separable over $F$.
5. If $K$ is a finite extension of $F$ and a minimal 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, if you 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 the fixed $\mathbb{Q}$, resulting in $\left\{ \mathbb{Q} ( \sqrt{2} , \sqrt{3} ) : \mathbb{Q} \right\} = 4$.

The reason a separable extension field is defined separately is because, generally, $\left\{ E : F \right\} \mid [ E : F ]$ holds, but it’s not always guaranteed to be the same.