# Field extension

Field extension

In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties. For instance, the set Q(√2) = {a + b√2 | a, bQ} is the smallest extension of Q which includes all solutions to the equation x2 = 2.

## Definitions

Let L be a field. If K is a subset of the underlying set of L which is closed with respect to the field operations and inverses in L, then K is said to be a subfield of L, and L is said to be an extension field of K. We then say that L /K, read as "L over K", is a field extension.

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 the field extension L /K.

Given a field extension L /K and a subset S of L, K(S) denotes the smallest subfield of L which contains K and S, a field generated by the adjunction of elements of S to K. If S consists of only one element s, K(s) is a shorthand for K({s}). A field extension of the form L = K(s) is called a simple extension and s is called a primitive element of the extension.

Given a field extension L /K, then L can also be considered as a vector space over K. The elements of L are the "vectors" and the elements of K are the "scalars", with vector addition and scalar multiplication obtained from the corresponding field operations. The dimension of this vector space is called the degree of the extension, and is denoted by [L : K].

An extension of degree 1 (that is, one where L is equal to K) is called a trivial extension. Extensions of degree 2 and 3 are called quadratic extensions and cubic extensions, respectively. Depending on whether the degree is finite or infinite the extension is called a finite extension or infinite extension.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Degree of a field extension — In mathematics, more specifically field theory, the degree of a field extension is a rough measure of the size of the extension. The concept plays an important role in many parts of mathematics, including algebra and number theory indeed in any… …   Wikipedia

• Dual basis in a field extension — In mathematics, the linear algebra concept of dual basis can be applied in the context of a finite extension L/K, by using the field trace. This requires the property that the field trace TrL/K provides a non degenerate quadratic form over K.… …   Wikipedia

• Field (mathematics) — This article is about fields in algebra. For fields in geometry, see Vector field. For other uses, see Field (disambiguation). In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it …   Wikipedia

• Extension (mathematics) — In mathematics, the word extension has many uses. See:Analysis* Carathéodory s extension theorem * Continuous linear extension * M. Riesz extension theorem * Krein extension theorem * Hahn Banach theoremAlgebra* Abelian extension * Algebraic… …   Wikipedia

• Field trace — In mathematics, the field trace is a linear mapping defined for certain field extensions. If L / K is a finite Galois extension, it is defined for α in L as the sum of all the conjugates: g (α)of α, for g in the Galois group G of L over K . It is …   Wikipedia

• Extension and contraction of ideals — In commutative algebra, the extension and contraction of ideals are operations performed on sets of ideals. Extension of an ideal Let A and B be two commutative rings with unity, and let f : A → B be a (unital) ring homomorphism. If mathfrak{a}… …   Wikipedia

• Extensión agraria — Saltar a navegación, búsqueda Extensión agraria en Laos, 2006 El concepto de Extensión agraria hace referencia a la aplicación de la investigación científica y los nuevos conocimientos a las prácticas agrarias a través de la educación a …   Wikipedia Español

• Glossary of field theory — Field theory is the branch of mathematics in which fields are studied. This is a glossary of some terms of the subject. (See field theory (physics) for the unrelated field theories in physics.) Definition of a field A field is a commutative ring… …   Wikipedia

• Field of definition — In mathematics, the field of definition of an algebraic variety V is essentially the smallest field to which the coefficients of the polynomials defining V can belong. Given polynomials, with coefficients in a field K , it may not be obvious… …   Wikipedia

• field — Synonyms and related words: DMZ, academic discipline, academic specialty, aceldama, achievement, acreage, aerodrome, agora, air base, airdrome, airfield, airport, alerion, ambit, amphitheater, ample scope, animal charge, annulet, answer,… …   Moby Thesaurus