Editing Field extension (section)

{{Use American English|date = January 2019}}
{{Short description|Construction of a larger algebraic field by "adding elements" to a smaller field}}
In [[mathematics]], particularly in [[algebra]], a '''field extension''' is a pair of [[Field (mathematics)|fields]] <math>K \subseteq L</math>, such that the operations of ''K'' are those of ''L'' [[Restriction (mathematics)|restricted]] to ''K''. In this case, ''L'' is an '''extension field''' of ''K'' and ''K'' is a '''subfield''' of ''L''.<ref>{{harvtxt|Fraleigh|1976|p=293}}</ref><ref>{{harvtxt|Herstein|1964|p=167}}</ref><ref>{{harvtxt|McCoy|1968|p=116}}</ref> For example, under the usual notions of [[addition]] and [[multiplication]], the [[complex number]]s are an extension field of the [[real number]]s; the real numbers are a subfield of the complex numbers.

Field extensions are fundamental in [[algebraic number theory]], and in the study of [[polynomial roots]] through [[Galois theory]], and are widely used in [[algebraic geometry]].