Definition:Separable Degree

Definition

Let $E / F$ be a field extension.

Definition 1

Let $S \subseteq E$ be the separable closure of $F$ in $E$.

The separable degree $\index E F_{\operatorname {sep} }$ of $E / F$ is the degree $\index S F$.

Definition 2

Let $\bar F$ be the algebraic closure of $F$.

The separable degree $\index E F_{\operatorname {sep} }$ of $E / F$ is the number of embeddings of $E$ into $\bar F$ that fix $F$.

Definition 3

Let $K$ be a normal extension of $F$ that contains $E$.

The separable degree $\index E F_{\operatorname {sep} }$ of $E / F$ is the number of embeddings of $E$ into $K$ that fix $F$.

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Also denoted as

The separable degree is also denoted $\index E F_s$.

Also see

• Definition:Inseparable Degree