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In mathematics, particularly in algebra, a field extension is a pair of fields
K
⊆
L
{\displaystyle K\subseteq L}
, such that the operations of K are those of L restricted to K. In this case, L is an extension field of K, and K is a subfield of L. For example, under the usual notions of addition and multiplication, the complex numbers are an extension field of the real numbers; 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.
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