nLab
algebraic number

Contents

Definition

An algebraic number is a root of a polynomial with integer coefficients (or, equivalently, with rational coeffients).

A number (especially a complex number) which is not algebraic is called transcendental; famous examples are the base (e=2.7\mathrm{e} = 2.7\ldots ) and period (2πi=6.28i2 \pi \mathrm{i} = 6.28\ldots \mathrm{i}, or equivalently π=3.14\pi = 3.14\ldots ) of the natural logarithm.

An algebraic integer is a root of a monic polynomial with integer coefficients. Given a field kk the (algebraic) number field K=k[P]K = k[P] over kk is the minimal field containing all the roots of a given polynomial PP with coefficients in kk. Usually one considers algebraic number fields over rational numbers.

Revised on July 27, 2011 18:18:11 by Toby Bartels (64.89.62.147)