Normal extension

This article defines a field extension property: a property that can be evaluated to true/false for any field extension.
View a complete list of field extension properties|View a complete list of field properties

Definition

Suppose $L$ is a field extension of a field $K$. In other words, $K$ is a subfield of $L$. We say that $L$ is a normal extension of $K$ if it satisfies the following equivalent conditions:

1. $L$ is an algebraic extension of $K$, with the property that for any $\alpha \in L$, the minimal polynomial of $\alpha$ over $K$ splits completely over $L$ (i.e., it can be written as a product of linear factors in $L[x]$).
2. $L$ is an algebraic extension of $K$, with the property that if $p(x)\in K[x]$ is an irreducible polynomial having a root in $L$, then $p(x)$ splits completely into linear factors in $L[x]$.
3. $L$ is the splitting field over $K$ of a (possibly infinite) collection of polynomials in $K[x]$. (When the collection of polynomials is finite, then we get the stronger notion of finite normal extension).

Relation with other properties

Stronger properties

• Galois extension
• Finite normal extension

Weaker properties

• Algebraic extension
• Extension for which every intermediate isomorphism extends to an automorphism
• Extension for which every intermediate automorphism extends to an automorphism
• Subnormal extension