Finite normal extension

This article defines a field extension property: a property that can be evaluated to true/false for any field extension.
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Definition

Suppose $L$ is a field extension of a field $K$. In other words, $K$ is a subfield of $L$. $L$ is termed a finite normal extension of $K$ if it is both a finite extension and a normal extension of $K$. More explicitly, it satisfies the following equivalent conditions:

1. $L$ is a finite extension of $K$, and if $p(x)\in K[x]$ is an irreducible polynomial having a root in $L$, then $p$ splits completely into linear factors over $L[x]$.
2. $L$ is a finite extension of $K$, and the minimal polynomial of any element of $L$ splits completely over $K$.
3. $L$ is the splitting field over $K$ of a finite collection of irreducible polynomials.
4. $L$ is the splitting field over $K$ of a single polynomial (not necessarily irreducible).
5. $L$ is the splitting field over $K$ of a single irreducible polynomial.

Relation with other properties

Stronger properties

• Finite Galois extension

Weaker properties

• Finite extension
• Normal extension