Rational Integer

Definition

The term rational integer is sometimes used to just refer to the set of integers Z to disambiguate from algebraic integers.

The term rational integer comes from the following result.

Lemma

The rational integers are exactly the only algebraic integers which are rational.

Proof

The only rational roots of a monic integer polynomial are integers. Thus every algebraic integer (root of a monic polynomial) must not be rational, or be an integer. For the other inclusion, we know that the minimal polynomial of any rational number pq is exactly X−pq and this number is an algebraic integer if and only if this polynomial has integer coefficients, that is if q=1.