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If F is the completion of an algebraic number field, its ring of integers is the completion of the latter's ring of integers. The ring of integers of an algebraic number field may be characterised as the elements which are integers in every non-archimedean completion. [3] For example, the p-adic integers Z p are the ring of integers of the p ...
The ring of algebraic integers is a Bézout domain, as a consequence of the principal ideal theorem. If the monic polynomial associated with an algebraic integer has constant term 1 or −1, then the reciprocal of that algebraic integer is also an algebraic integer, and each is a unit , an element of the group of units of the ring of algebraic ...
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers.
It follows that the algebraic integers in form a ring denoted called the ring of integers of . 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 .
An ideal in the ring of integers of an algebraic number field is principal if it consists of multiples of a single element of the ring, and nonprincipal otherwise. By the principal ideal theorem any nonprincipal ideal becomes principal when extended to an ideal of the Hilbert class field .
The algebraic integers in a finite extension field k of the rationals Q form a subring of k, called the ring of integers of k, a central object of study in algebraic number theory. In this article, the term ring will be understood to mean commutative ring with a multiplicative identity.
In mathematics, Dirichlet's unit theorem is a basic result in algebraic number theory due to Peter Gustav Lejeune Dirichlet. [1] It determines the rank of the group of units in the ring O K of algebraic integers of a number field K. The regulator is a positive real number that determines how "dense" the units are.
In algebraic number theory, R will be taken to be the ring of integers, which is Dedekind and thus regular. It follows that Pic(R) is a finite group (finiteness of class number) that measures the deviation of the ring of integers from being a PID.