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 is a field extension of a field (i.e., is a subfield of ). We say that is a finite extension of if it satisfies the following equivalent conditions:
- The degree of the extension, denoted , is finite, where the degree is defined as the dimension of as a -vector space.
- is generated over by a finite collection of elements, each of which is algebraic over .
- is an algebraic extension of that is also finitely generated.
- is a simple extension of that is also algebraic.
- There is an irreducible polynomial and an isomorphism , such that the isomorphism restricts to the identity on .