# 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.

Algebraic extensions form a *distinguished class* : that is, they have the properties (i) for $M / L / K$ we have $M/L,\,L/K$ algebraic if and only if $M/K$ is algebraic; (ii) $M / K,\,L/K $ algebraic implies $ ML/L$ algebraic.

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 Zbl 0768.12001 |

[b2] | Steven Roman, Field Theory, Graduate Texts in Mathematics 158 (2nd edition) Springer (2007) ISBN 0-387-27678-5 Zbl 1172.12001 |

**How to Cite This Entry:**

Algebraic extension.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Algebraic_extension&oldid=54399