# Algebraic element

In

mathematics, the

rootss 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).

- 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**.

## Properties

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

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.