Algebraic element

In mathematics, the root (mathematics)s of polynomials are in abstract algebra called algebraic elements. They can be created in a larger structure ( adjoined ), not simply found to exist in a given one.

More precisely, if L is a field extension 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 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 two is algebraic over Q, since it is the root of the polynomial g ( x ) = x 2 - 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 a vector space 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 field. The field of complex numbers is an example.