In mathematics, an algebraic integer is a complex number α that is a root of an equation

P(x) = 0

where P(x) is a monic polynomial with integer coefficients.

The algebraic integers are all therefore algebraic numbers, but not all algebraic numbers are algebraic integers.

One may show that if P(x) is a non-monic primitive polynomial with integer coefficients that is irreducible over Q, then none of the roots of P are algebraic integers. Here the word primitive means that coefficients of P are relatively prime (i.e. the greatest common divisor of the set of coefficients of P is 1; note that this is weaker than requiring the coefficients to be pairwise relatively prime.)

The sum of two algebraic integers is an algebraic integer, and so is their difference; their product is too, but not necessarily their ratio. An integer root of an algebraic integer is also an algebraic integer. So all radical integer s are algebraic integers but not all algebraic integers are radical integers. In other words, the algebraic integers form a ring that is closed under the operation of extraction of roots.

The algebraic integers are a Bezout domain .