Euclidean Rings of Algebraic Integers

The Ring of Integers, Euclidean Rings and Modulo Integers

The binary operation multint on Z is defined as follows: (Def. 1) For all elements a, b of Z holds (multint)(a, b) = ·R(a, b). The unary operation compint on Z is defined as follows: (Def. 2) For every element a of Z holds (compint)(a) = −R(a). The double loop structure INT.Ring is defined by: (Def. 3) INT.Ring = 〈Z, +Z,multint, 1(∈ Z), 0(∈ Z)〉. Let us mention that INT.Ring is strict and non em...

Primitive Recursive Decidability for Large Rings of Algebraic Integers

Algebraic Numbers and Algebraic Integers

c © W W L Chen, 1984, 2013. This chapter originates from material used by the author at Imperial College London between 1981 and 1990. It is available free to all individuals, on the understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied, with or without permission from the author. However, this document may not be kept on any information storage an...

About Euclidean Rings

In this article all rings are commutative with unit, all modules are unitary. Given a ring A, its multiplicative group of units (i.e. invertible elements) is denoted by A*. The customary definition of a Euclidean ring is that it is a domain A together with a map F : A + N (the nonnegative integers) such that (1) I : p(a) for a, b E r3 (0); (2) given a, b E -‘-I, b m;’ 0, there exist q and Y in ...

Approximating Rings of Integers in Number Elds Approximating Rings of Integers in Number Elds

In this paper we study the algorithmic problem of nding the ring of integers of a given algebraic number eld. In practice, this problem is often considered to be well-solved, but theoretical results indicate that it is intractable for number elds that are deened by equations with very large coeecients. Such elds occur in the number eld sieve algorithm for factoring integers. Applying a variant ...