Algebraic integer

In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients are integers. The set of all algebraic integers A is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers. The ring of integers of a number field K, denoted by OK, is the intersection of K and A: it can also be characterised as the maximal order of the field K. Each algebraic integer belongs to the ring of integers of some number field. A number α is an algebraic integer if and only if the ring is finitely generated as an abelian group, which is to say, as a -module. The following are equivalent definitions of an algebraic integer. Let K be a number field (i.e., a finite extension of , the field of rational numbers), in other words, for some algebraic number by the primitive element theorem. α ∈ K is an algebraic integer if there exists a monic polynomial such that f(α) = 0. α ∈ K is an algebraic integer if the minimal monic polynomial of α over is in . α ∈ K is an algebraic integer if is a finitely generated -module. α ∈ K is an algebraic integer if there exists a non-zero finitely generated -submodule such that αM ⊆ M. Algebraic integers are a special case of integral elements of a ring extension. In particular, an algebraic integer is an integral element of a finite extension . The only algebraic integers which are found in the set of rational numbers are the integers. In other words, the intersection of and A is exactly . The rational number a/b is not an algebraic integer unless b divides a. Note that the leading coefficient of the polynomial bx − a is the integer b. As another special case, the square root of a nonnegative integer n is an algebraic integer, but is irrational unless n is a perfect square. If d is a square-free integer then the extension is a quadratic field of rational numbers.