- Algebraic extension
In
abstract algebra , afield extension "L" /"K" is called algebraic if every element of "L" is algebraic over "K", i.e. if every element of "L" is a root of some non-zeropolynomial with coefficients in "K". Field extensions which are not algebraic, i.e. which containtranscendental element s, are called transcendental.For example, the field extension R/Q, that is the field of
real number s as an extension of the field ofrational number s, is transcendental, while the field extensions C/R and Q(√2)/Q are algebraic, where C is the field ofcomplex number s.All transcendental extensions are of infinite degree. This in turn implies that all finite extensions are algebraic. The converse is not true however: there are infinite extensions which are algebraic. For instance, the field of all
algebraic number s is an infinite algebraic extension of the rational numbers.If "a" is algebraic over "K", then "K" ["a"] , the set of all polynomials in "a" with coefficients in "K", is not only a ring but a field: an algebraic extension of "K" which has finite degree over "K". In the special case where "K"=Q is the field of rational numbers, Q ["a"] is an example of an
algebraic number field .A field with no proper algebraic extensions is called algebraically closed. An example is the field of
complex number s. Every field has an algebraic extension which is algebraically closed (called itsalgebraic closure ), but proving this in general requires some form of theaxiom of choice .An extension "L"/"K" is algebraic
if and only if every sub "K"-algebra of "L" is a field.Generalizations
Model theory generalizes the notion of algebraic extension to arbitrary theories: an embedding of "M" into "N" is called an algebraic extension if for every "x" in "N" there is a formula "p" with parameters in "M", such that "p"("x") is true and the set:{"y" in "N" | "p"("y")}
is finite. It turns out that applying this definition to the theory of fields gives the usual definition of algebraic extension. The
Galois group of "N" over "M" can again be defined as the group of automorphisms, and it turns out that most of the theory of Galois groups can be developed for the general case.See also
*
Algebraically closed field
*Algebraic closure References
* S. Lang, "Algebra (3 ed)",
Addison-Wesley , 1993, ISBN 0-201-55540-9. Chap.V.1, p.223.
* P.J. McCarthy, "Algebraic extensions of fields",Dover Publications , 1991, ISBN 0-486-66651-4.
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