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 (that is, the coefficient of the largest power of x in P(x) is one) with integer coefficients.

All algebraic integers are therefore algebraic numbers, but it can be shown that 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 coprime (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 integers 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 .