In mathematics, an algebraic integer is an algebraic number that is a root (mathematics) of a monic polynomial (i.e. a polynomial whose leading coefficient is 1) with integer coefficients. Examples include the Gaussian integers and Eisenstein integers.

One may show that if P ( x ) is a non-monic primitive polynomial with integer coefficients that is irreducible polynomial 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_(mathematics) that is closed under the operation of extraction of roots.

The algebraic integers are a Bézout domain.

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