Definition:Normal Extension

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Definition

Definition 1

Let $L / K$ be a field extension.

Then $L / K$ is a normal extension if and only if:

for every irreducible polynomial $f \in K \left[{x}\right]$ with at least one root in $L$, $f$ splits completely in $L$.


Definition 2

Let $L / K$ be a field extension.

Let $\overline K$ be the algebraic closure of $K$.

Let $\operatorname{Gal} \left({L / K}\right)$ denote the set of embeddings of $L$ in $\overline K$ which fix $K$ pointwise.


Then $L/K$ is a normal extension if and only if:

$\sigma \left({L}\right) = L$

for each $\sigma \in \operatorname{Gal} \left({L / K}\right)$.


Also known as

A normal extension is also known as a quasi-Galois extension.


Also see


Note

The second definition captures the core concept of a normal extension as a field extension in which the embeddings are actually field automorphisms.

This, in turn, endows $\operatorname{Gal} \left({L / K}\right)$ with a group structure since we can now compose embeddings.

However, in developing the theory, the first definition tends to be more useful.