nLab
algebraic number

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 ($\mathrm{e}=2.7\dots$) and period ($2\pi \mathrm{i}=6.28\dots \mathrm{i}$, or equivalently $\pi =3.14\dots$) of the natural logarithm.

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

Related concepts

• algebraic number theory

  • algebraic number, algebraic integer

  • number field