Algebraic extension

2020 Mathematics Subject Classification: Primary: 12F [MSN][ZBL]

A field extension $K/k$ in which every element of $K$ is algebraic over $k$; that is, every element of $K$ is the root of a non-zero polynomial with coefficients in $k$. A finite degree extension is necessarily algebraic, but the converse does not hold: for example, the field of algebraic numbers, the algebraic closure of the field of rational numbers, is an algebraic extension but not of finite degree.

An extension which is not algebraic is a transcendental extension.

References

[b1]	Paul J. McCarthy, "Algebraic Extensions of Fields", Courier Dover Publications (2014) ISBN 048678147X