Algebraic number
 

is any complex number (including real numbers) that is a root of a non-zero polynomial
(that is, a value which causes the polynomial to equal 0) in one variable with
rational coefficients (or equivalently, by clearing denominators, with integer coefficients).

All integers and rational numbers are algebraic, as are all roots of integers.

Real and complex numbers that are not algebraic, such as π and e, are called
transcendental numbers.

The set of complex numbers is uncountable,
but the set of algebraic numbers is countable and has measure zero
in the Lebesgue measure as a subset of the complex numbers.

In that sense, almost all complex numbers are transcendental.

In mathematics,
an algebraic number field (or simply number field) F is a finite degree
(and hence algebraic) field extension of the field of rational numbers Q.
Thus F is a field that contains Q and has finite dimension when
considered as a vector space over Q.

The study of algebraic number fields, and, more generally,
of algebraic extensions of the field of rational numbers,
is the central topic of algebraic number theory.