algebraic integer

Among all algebraic numbers, some deserve to be called “integers” in the same spirit that 5 is an integer but 5/2 is not. The honest test is to demand that the cleanest possible polynomial equation already has whole-number coefficients and a leading coefficient of 1, with no denominators forced upon you. Such numbers behave like the integers do inside the rationals: they are closed under addition and multiplication and form a ring.

Formally, a complex number a is an algebraic integer if it is a root of a monic polynomial with integer coefficients — that is, x^n + c_{n-1} x^{n-1} + ... + c_0 with each c_i in Z. Equivalently, a is an algebraic integer exactly when its minimal polynomial over the rationals has integer coefficients. A rational number is an algebraic integer if and only if it is an ordinary integer (this is the rational root theorem in disguise).

The algebraic integers form a ring; intersected with any number field K they give the ring of integers of K, the central object of the subject. Algebraic integers are exactly the algebraic numbers that are integral over Z, so they are the number-theoretic instance of an integral extension.