# Algebraic element

In mathematics, 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 2 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**: 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

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 nonzero 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.

## See also

## References

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