Field extension

In mathematics, particularly in algebra, a field extension (denoted ${\displaystyle L/K}$) is a pair of fields ${\displaystyle K\subseteq L}$, 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.^[1][2][3] 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.