Editing Algebraic extension (section)

{{Short description|Extension of a mathematical field with polynomial roots}}
{{use mdy dates|date=September 2021}}
{{Use American English|date = January 2019}}

In [[mathematics]], an '''algebraic extension''' is a [[field extension]] {{math|''L''/''K''}} such that every element of the larger [[field (mathematics)|field]] {{mvar|L}} is [[algebraic element|algebraic]] over the smaller field {{mvar|K}}; that is, every element of {{mvar|L}} is a root of a non-zero [[polynomial]] with coefficients in {{mvar|K}}.<ref>Fraleigh (2014), Definition 31.1, p. 283.</ref><ref>Malik, Mordeson, Sen (1997), Definition 21.1.23, p. 453.</ref> A field extension that is not algebraic, is said to be [[Field extension#Transcendental extension|transcendental]], and must contain [[transcendental element]]s, that is, elements that are not algebraic.<ref>Fraleigh (2014), Definition 29.6, p. 267.</ref><ref>Malik, Mordeson, Sen (1997), Theorem 21.1.8, p. 447.</ref>

The algebraic extensions of the field <math>\Q</math> of the [[rational number]]s are called [[algebraic number field]]s and are the main objects of study of [[algebraic number theory]]. Another example of a common algebraic extension is the extension <math>\Complex/\R</math> of the [[real number]]s by the [[complex number]]s.