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.

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Related concepts (40)
Ring (mathematics)
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. In other words, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.
Finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod p when p is a prime number. The order of a finite field is its number of elements, which is either a prime number or a prime power.
Field (mathematics)
In mathematics, a field is a set on which addition, subtraction, multiplication, and division are defined and behave as the corresponding operations on rational and real numbers do. A field is thus a fundamental algebraic structure which is widely used in algebra, number theory, and many other areas of mathematics. The best known fields are the field of rational numbers, the field of real numbers and the field of complex numbers.
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