Algebraic element

Summary

In mathematics, if L is an extension field of K, then an element a of L is called an algebraic element over K, or just algebraic over K, if there exists some non-zero polynomial g(x) with coefficients in K such that g(a) = 0. Elements of L that are not algebraic over K are called transcendental over K.

These notions generalize the algebraic numbers and the transcendental numbers (where the field extension is C/Q, with C being the field of complex numbers and Q being the field of rational numbers).

Examples

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  • The square root of 2 is algebraic over Q, since it is the root of the polynomial g(x) = x2 − 2 whose coefficients are rational.
  • Pi is transcendental over Q but algebraic over the field of real numbers R: it is the root of g(x) = x − π, whose coefficients (1 and −π) are both real, but not of any polynomial with only rational coefficients. (The definition of the term transcendental number uses C/Q, not C/R.)

Properties

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The following conditions are equivalent for an element   of  :

  •   is algebraic over  ,
  • the field extension   is algebraic, i.e. every element of   is algebraic over   (here   denotes the smallest subfield of   containing   and  ),
  • the field extension   has finite degree, i.e. the dimension of   as a  -vector space is finite,
  •  , where   is the set of all elements of   that can be written in the form   with a polynomial   whose coefficients lie in  .

To make this more explicit, consider the polynomial evaluation  . This is a homomorphism and its kernel is  . If   is algebraic, this ideal contains non-zero polynomials, but as   is a euclidean domain, it contains a unique polynomial   with minimal degree and leading coefficient  , which then also generates the ideal and must be irreducible. The polynomial   is called the minimal polynomial of   and it encodes many important properties of  . Hence the ring isomorphism   obtained by the homomorphism theorem is an isomorphism of fields, where we can then observe that  . Otherwise,   is injective and hence we obtain a field isomorphism  , where   is the field of fractions of  , i.e. the field of rational functions on  , by the universal property of the field of fractions. We can conclude that in any case, we find an isomorphism   or  . Investigating this construction yields the desired results.

This characterization can be used to show that the sum, difference, product and quotient of algebraic elements over   are again algebraic over  . For if   and   are both algebraic, then   is finite. As it contains the aforementioned combinations of   and  , adjoining one of them to   also yields a finite extension, and therefore these elements are algebraic as well. Thus set of all elements of   that are algebraic over   is a field that sits in between   and  .

Fields that do not allow any algebraic elements over them (except their own elements) are called algebraically closed. The field of complex numbers is an example. If   is algebraically closed, then the field of algebraic elements of   over   is algebraically closed, which can again be directly shown using the characterisation of simple algebraic extensions above. An example for this is the field of algebraic numbers.

See also

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References

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  • Lang, Serge (2002), Algebra, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001