Field extension

In mathematics, particularly in algebra, a field extension is a pair of fields 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 of a field is a subset that is a field with respect to the field operations inherited from . Equivalently, a subfield is a subset that contains , and is closed under the operations of addition, subtraction, multiplication, and taking the inverse of a nonzero element of . As 1 – 1 = 0, the latter definition implies and 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 . The characteristic of a subfield is the same as the characteristic of the larger field. If K is a subfield of L, then L is an extension field or simply extension of K, and this pair of fields is a field extension. Such a field extension is denoted L / K (read as "L over K"). If L is an extension of F, which is in turn an extension of K, then F is said to be an intermediate field (or intermediate extension or subextension) of L / K. Given a field extension L / K, the larger field L is a K-vector space. The dimension of this vector space is called the degree of the extension and is denoted by [L : K]. The degree of an extension is 1 if and only if the two fields are equal. In this case, the extension is a trivial extension. Extensions of degree 2 and 3 are called quadratic extensions and cubic extensions, respectively. A finite extension is an extension that has a finite degree.