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Algebraic element

In abstract algebra, the roots of polynomials are called algebraic elements. More precisely, if L/K is a field extension then aL is an algebraic element over K, or algebraic over K for short, iff there exists some non-zero polynomial g(x) with coefficients in K such that g(a)=0. Elements of L which 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, 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.

Properties

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

  • a is algebraic over K
  • the field extension K(a)/K has finite degree, i.e. the dimension of K(a) as a K-vector space is finite. (Here K(a) denotes the smallest subfield of L containing K and a)
  • 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.

This characterization can be used to show that the sum, difference, product and quotient of algebraic elements over K are again algebraic over K. The set of all elements of L which are algebraic over K is a field that sits in between L and K.

If a is algebraic over K, then there are many non-zero polynomials g(x) with coefficients in K such that g(a) = 0. However there is a single one with smallest degree and with leading coefficient 1. This is the minimal polynomial of a and it encodes many important properties of a.

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.



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