Similarly, one may define the ring of integers as the subset of where are restricted to be integers.
12.
We may define the Frobenius map for elements of the ring of integers of as an automorphism of such that
13.
By the principal ideal theorem, every prime ideal of generates a principal ideal of the ring of integers of.
14.
The problem for the ring of integers of algebraic number fields other than those covered by the results above remains open.
15.
For example, the dyadic fractions are the localization of the ring of integers with respect to the powers of two.
16.
More formally, the-adic expansions can be used to define the ring of integers modulo which is also sometimes written.
17.
Strictly speaking it is the ring of integers that is Euclidean since fields are trivially Euclidean domains, but the terminology is standard.
18.
By the Chinese remainder theorem, this ring factors into the direct product of rings of integers mod " p ".
19.
Using this and denoting repeated addition by a multiplication by a positive integer allows identifying abelian groups with modules over the ring of integers.
20.
In the case of the ring of integers \ mathbb Z, the Lasker Noether theorem is equivalent to the fundamental theorem of arithmetic.
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