Definition:Normal Extension
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 $f \in K \sqbrk x$ 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 $\Gal {L / K}$ 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:
- $\sigma \sqbrk L = L$
for each $\sigma \in \Gal {L / K}$.
Also known as
A normal extension is also known as a quasi-Galois extension.
Also see
- Equivalence of Definitions of Normal Extension
- Definition:Separable Extension
- Definition:Galois Extension
- Results about normal extensions can be found here.
Note
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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 $\Gal {L / K}$ with a group structure since we can now compose embeddings.
However, in developing the theory, the first definition tends to be more useful.
Sources
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- This article incorporates material from normal extension on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.