Field extension

In abstract algebra, an extension of a field K is a field L which contains K as a subfield.

Related Topics:
Abstract algebra - Field - Subfield

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For example, C (the field of complex numbers)

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is an extension of R (the field of real numbers),

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and R is itself an extension of Q

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(the field of rational numbers).

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The notation L/K is often used to denote the fact that L is an extension of K.

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More generally, an extension of K is a separate pair of fields K* and L, where L contains K* as a subfield, and K is isomorphic to K*. Where it does not cause confusion, we identify K and K*, as above and below.

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Given a field extension L/K,

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L can be considered as a vector space over K,

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with vector addition being the field addition on L,

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and scalar multiplication being a restriction of the field multiplication on L.

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The dimension of this vector space

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is called the degree of the extension, and is denoted .

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The extension is said to be finite or infinite according as

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the degree is finite or infinite.

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For example, = 2, so this extension is finite.

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By contrast, = c (the cardinality of the continuum),

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so this extension is infinite.

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If M is an extension of L which is an extension of K,

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then = ..

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If L is an extension of K, then an element of L which is a root

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of a nonzero polynomial over K is said to be algebraic over K.

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If it is not algebraic then it is said to be transcendental.

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(The special case where L = C and K = Q is particularly important. see algebraic number and transcendental number.)

Related Topics:
Algebraic number - Transcendental number

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If every element of L is algebraic over K,

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then the extension L/K is said to be algebraic,

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otherwise it is said to be transcendental.

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If every element of L  K is transcendental over K,

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then the extension is said to be pure transcendental.

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It can be shown that an extension is algebraic if and only if it is the

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union of its finite subextensions.

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In particular, every finite extension is algebraic.

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For example, C/R, being finite, is algebraic.

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But R/Q is transcendental, although not pure transcendental.

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See algebraic extension for more information on algebraic extensions.

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If L/K is a field extension and V is a subset of L, then the field K(V) is defined to be the smallest subfield of L which contains K and V. It consists of all those elements of L which can be gotten using a finite number of field operations +, -, *, / applied to elements from K and V. If L = K(V), we say that L is generated by V.

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A field extension generated by a single element is called a simple extension.

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A simple extension is finite if generated by an algebraic element,

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and pure transcendental if generated by a transcendental element.

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For example, C is a simple extension of R, as it is generated by i (the square root of minus one).

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The extension R/Q is not simple, as it is neither finite nor pure transcendental.

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A field extension which has a Galois group is called a Galois extension.

Related Topics:
Galois group - Galois extension

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If the Galois group is Abelian, then the extension is called an Abelian extension.

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For example, C/R is a Abelian extension, its Galois group being of order 2.

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But R/Q is not a Galois extension, since, for example, the polynomial x3 − 2, while having a root in R, does not split over R.

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