Definition 9.28.1. Algebraic field extensions.

A field extension $K/k$ is called

*algebraic*if every element of $K$ is algebraic over $k$.An algebraic extension $k'/k$ is called

*separable*if every $\alpha \in k'$ is separable over $k$.An algebraic extension $k'/k$ is called

*purely inseparable*if the characteristic of $k$ is $p > 0$ and for every element $\alpha \in k'$ there exists a power $q$ of $p$ such that $\alpha ^ q \in k$.An algebraic extension $k'/k$ is called

*normal*if for every $\alpha \in k'$ the minimal polynomial $P(T) \in k[T]$ of $\alpha $ over $k$ splits completely into linear factors over $k'$.An algebraic extension $k'/k$ is called

*Galois*if it is separable and normal.

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