information:classification
Algebraicity, and ring of integers

Generally, in abstract algebra, a field extension K / L {\displaystyle K/L} is algebraic if every element f {\displaystyle f} of the bigger field K {\displaystyle K} is the zero of a (nonzero) polynomial with coefficients e 0, …, e m {\displaystyle e_{0},\ldots,e_{m}} in L {\displaystyle L}:

Every field extension of finite degree is algebraic. (Proof: for x {\displaystyle x} in K {\displaystyle K}, simply consider 1, x, x 2, x 3, … {\displaystyle 1,x,x^{2},x^{3},\ldots } – we get a linear dependence, i.e. a polynomial that x {\displaystyle x} is a root of.) In particular this applies to algebraic number fields, so any element f {\displaystyle f} of an algebraic number field K {\displaystyle K} can be written as a zero of a polynomial with rational coefficients. Therefore, elements of K {\displaystyle K} are also referred to as algebraic numbers. Given a polynomial p {\displaystyle p} such that p (f) = 0 {\displaystyle p(f)=0}, it can be arranged such that the leading coefficient e m {\displaystyle e_{m}} is one, by dividing all coefficients by it, if necessary. A polynomial with this property is known as a monic polynomial. In general it will have rational coefficients.

If, however, the monic polynomial's coefficients are actually all integers, f {\displaystyle f} is called an algebraic integer.

Any (usual) integer z ∈ Z {\displaystyle z\in \mathbb {Z} } is an algebraic integer, as it is the zero of the linear monic polynomial:

It can be shown that any algebraic integer that is also a rational number must actually be an integer, hence the name "algebraic integer". Again using abstract algebra, specifically the notion of a finitely generated module, it can be shown that the sum and the product of any two algebraic integers is still an algebraic integer. It follows that the algebraic integers in K {\displaystyle K} form a ring denoted O K {\displaystyle {\mathcal {O}}_{K}} called the ring of integers of K {\displaystyle K}. It is a subring of (that is, a ring contained in) K {\displaystyle K}. A field contains no zero divisors and this property is inherited by any subring, so the ring of integers of K {\displaystyle K} is an integral domain. The field K {\displaystyle K} is the field of fractions of the integral domain O K {\displaystyle {\mathcal {O}}_{K}}. This way one can get back and forth between the algebraic number field K {\displaystyle K} and its ring of integers O K {\displaystyle {\mathcal {O}}_{K}}. Rings of algebraic integers have three distinctive properties: firstly, O K {\displaystyle {\mathcal {O}}_{K}} is an integral domain that is integrally closed in its field of fractions K {\displaystyle K}. Secondly, O K {\displaystyle {\mathcal {O}}_{K}} is a Noetherian ring. Finally, every nonzero prime ideal of O K {\displaystyle {\mathcal {O}}_{K}} is maximal or, equivalently, the Krull dimension of this ring is one. An abstract commutative ring with these three properties is called a Dedekind ring (or Dedekind domain), in honor of Richard Dedekind, who undertook a deep study of rings of algebraic integers.