algebraic extension

Definition 1.

Let L/K be an extension of fields. L/K is said to be an algebraic extensionMathworldPlanetmath of fields if every element of L is algebraic over K. If L/K is not algebraic then we say that it is a transcendental extension of fields.


  1. 1.

    Let L=(2). The extension L/ is an algebraic extension. Indeed, any element αL is of the form


    for some q,t. Then αL is a root of

  2. 2.

    The field extension / is not an algebraic extension. For example, π is a transcendental numberMathworldPlanetmath over (see pi). So / is a transcendental extension of fields.

  3. 3.

    Let K be a field and denote by K¯ the algebraic closureMathworldPlanetmath of K. Then the extension K¯/K is algebraic.

  4. 4.

    In general, a finite extensionMathworldPlanetmath of fields is an algebraic extension. However, the converse is not true. The extension ¯/ is far from finite.

  5. 5.

    The extension (π)/ is transcendental because π is a transcendental number, i.e. π is not the root of any polynomialPlanetmathPlanetmath p(x)[x].

Title algebraic extension
Canonical name AlgebraicExtension
Date of creation 2013-03-22 13:57:27
Last modified on 2013-03-22 13:57:27
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 7
Author alozano (2414)
Entry type Definition
Classification msc 12F05
Synonym algebraic field extension
Related topic Algebraic
Related topic FiniteExtension
Related topic AFiniteExtensionOfFieldsIsAnAlgebraicExtension
Related topic ProofOfTranscendentalRootTheorem
Related topic EquivalentConditionsForNormalityOfAFieldExtension
Defines examples of field extension
Defines transcendental extension