In mathematics, particularly in algebra, a field extension 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. 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.

A subfield ${\displaystyle K}$ of a field ${\displaystyle L}$ is a subset ${\displaystyle K\subseteq L}$ that is a field with respect to the field operations inherited from ${\displaystyle L}$. Equivalently, a subfield is a subset that contains the multiplicative identity ${\displaystyle 1}$, and is closed under the operations of addition, subtraction, multiplication, and taking the inverse of a nonzero element of ${\displaystyle K}$.

As 1 – 1 = 0, the latter definition implies ${\displaystyle K}$ and ${\displaystyle L}$ have the same zero element.

For example, the field of rational numbers is a subfield of the real numbers, which is itself a subfield of the complex numbers. More generally, the field of rational numbers is (or is isomorphic to) a subfield of any field of characteristic ${\displaystyle 0}$.

The characteristic of a subfield is the same as the characteristic of the larger field.

If ${\displaystyle K}$ is a subfield of ${\displaystyle L}$, then ${\displaystyle L}$ is an extension field or simply extension of ${\displaystyle K}$, and this pair of fields is a field extension. Such a field extension is denoted ${\displaystyle L/K}$ (read as "${\displaystyle L}$ over ${\displaystyle K}$").

If ${\displaystyle L}$ is an extension of ${\displaystyle F}$, which is in turn an extension of ${\displaystyle K}$, then ${\displaystyle F}$ is said to be an intermediate field (or intermediate extension or subextension) of ${\displaystyle L/K}$.