Field extension

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In mathematics, more specifically 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.

Contents

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 by definition K is a subfield of L, and L is an extension field of K. 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 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.

Notes

The notation L/K is purely formal and does not imply the formation of a quotient ring or quotient group or any other kind of division. In some literature the notation L:K is used.

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