The Stacks project

Definition 9.28.1. Algebraic field extensions.

  1. A field extension $K/k$ is called algebraic if every element of $K$ is algebraic over $k$.

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

  3. 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 $\alpha ^ q \in k$.

  4. 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 $k'$.

  5. An algebraic extension $k'/k$ is called Galois if it is separable and normal.

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View Definition 9.28.1 as pdf