The arithmetic of number rings

Summary

We describe the main structural results on number rings, that is, integral domains for which the field of fractions is a number field. Whenever possible, we avoid the algorithmically undesirable hypothesis that the number ring in question is integrally closed.

The ring ℤ of ‘ordinary’ integers lies at the very root of number theory, and when studying its properties, the concept of divisibilityof integers naturally leads to basic notions as primality and congruences. By the ‘fundamental theorem of arithmetic’, ℤ admits unique prime factor decompositionof nonzero integers. Though one may be inclined to take this theorem for granted, its proof is not completely trivial: it usually employs the Euclidean algorithm to show that the prime numbers, which are defined as irreducibleelements having only ‘trivial’ divisors, are prime elementsthat only divide a product of integers if they divide one of the factors.