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On the other hand, the ring of integers in a number field is always a Dedekind domain.

Generalizing this simple result to more general rings of integers is a basic problem in algebraic number theory.

In fact, every ideal of the ring of integers is principal.

Its competition is then called the ring of p-adic integers.

The rings of integers in number fields are Dedekind domains.

Yet more general is the class of rings of algebraic integers.

By contrast, the ring of integers is not infinitely divisible.

Therefore, the ring of integers of F is an integral domain.

The problem for the ring of integers of algebraic number fields other than those covered by the results above remains open.

We let Z be the ring of p-adic integers.