Finite 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

Definition with symbols

Suppose $L$ is a field extension of a field $K$ (i.e., $K$ is a subfield of $L$). We say that $L$ is a finite extension of $K$ if it satisfies the following equivalent conditions:

1. The degree of the extension, denoted $[L:K]$, is finite, where the degree is defined as the dimension of $L$ as a $K$-vector space.
2. $L$ is generated over $K$ by a finite collection of elements, each of which is algebraic over $K$.
3. $L$ is an algebraic extension of $K$ that is also finitely generated.
4. $L$ is a simple extension of $K$ that is also algebraic.
5. There is an irreducible polynomial $p(x)\in K[x]$ and an isomorphism $L\cong K[x]/(p(x))$, such that the isomorphism restricts to the identity on $K$.

Relation with other properties

Stronger properties

• Finite normal extension
• Finite separable extension
• Finite Galois extension

Weaker properties

• Algebraic extension
• Finitely generated extension