Algebraic element


In mathematics, if is an associative algebra over, then an element of is an algebraic element over, or just algebraic over, if there exists some non-zero polynomial with coefficients in such that. Elements of that are not algebraic over are transcendental over. A special case of an associative algebra over is an extension field of.
These notions generalize the algebraic numbers and the transcendental numbers.

Examples

  • The square root of 2 is algebraic over, since it is the root of the polynomial whose coefficients are rational.
  • Pi is transcendental over but algebraic over the field of real numbers : it is the root of, whose coefficients are both real, but not of any polynomial with only rational coefficients.

Properties

The following conditions are equivalent for an element of an extension field of :
  • is algebraic over,
  • the field extension is algebraic, i.e. every element of is algebraic over,
  • 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 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.