Algebraic integer

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In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some polynomial which is monic (its leading coefficient is 1) and has coefficients that are all 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 characterized 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 Z [ α ] {\displaystyle \mathbb {Z} [\alpha ]} is finitely generated as an abelian group, which is to say, as a Z {\displaystyle \mathbb {Z} }...

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