In mathematics, if A is an associative algebra over K, then an element a of A is an algebraic element over K, or just algebraic over K, if there exists some non-zero polynomial g(x) \in K[x] with coefficients in K such that g(a) = 0.[1] Elements of A that are not algebraic over K are transcendental over K. A special case of an associative algebra over K is an extension field L of K. If all elements of L are algebraic over K, then L/K is called an algebraic extension.

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

  • 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.)
  • An indeterminate x in a field K(x) of rational functions or K((x)) of formal Laurent series is defined to be transcendental over K.
  • Algebraic functions are the algebraic elements over a field of rational functions K(x1, ..., xm).

Properties

The following conditions are equivalent for an element a of an extension field L of K:

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

To make this more explicit, consider the polynomial evaluation \varepsilon_a: K[X] \rightarrow K(a),\, P \mapsto P(a). This is a homomorphism and its kernel is \{P \in K[X] \mid P(a) = 0 \}. If a is algebraic, this ideal contains non-zero polynomials, but as K[X] is a euclidean domain, it contains a unique polynomial p with minimal degree and leading coefficient 1, which then also generates the ideal and must be irreducible. The polynomial p is called the minimal polynomial of a and it encodes many important properties of a. Hence the ring isomorphism K[X]/(p) \rightarrow \mathrm{im}(\varepsilon_a) obtained by the homomorphism theorem is an isomorphism of fields, where we can then observe that \mathrm{im}(\varepsilon_a) = K(a). Otherwise, \varepsilon_a is injective and hence we obtain a field isomorphism K(X) \rightarrow K(a), where K(X) is the field of fractions of K[X], i.e. the field of rational functions on K, by the universal property of the field of fractions. We can conclude that in any case, we find an isomorphism K(a) \cong K[X]/(p) or K(a) \cong K(X). 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 K are again algebraic over K. For if a and b are both algebraic, then (K(a))(b) is finite. As it contains the aforementioned combinations of a and b, adjoining one of them to K also yields a finite extension, and therefore these elements are algebraic as well. Thus set of all elements of L that are algebraic over K is a field that sits in between L and K.

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 L is algebraically closed, then the field of algebraic elements of L over K 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

References

  1. ^ Roman, Steven (2008). "18". Advanced Linear Algebra. Graduate Texts in Mathematics. New York, NY: Springer New York Springer e-books. pp. 458–459. ISBN 978-0-387-72831-5.