Taking " R " = " Z " this construction tells us precisely that rings of integers of number fields are Dedekind domains.
32.
We obtain an element of the ring of integers in this way; this is a polynomial of degree four in with coefficients in the-adic integers.
33.
In the general case, by Dirichlet's unit theorem, the group of units in the ring of integers of " K " is infinite.
34.
And back, starting from an algebraic number field ( an extension of rational numbers ), its ring of integers can be extracted, which includes as its subring.
35.
The " maximal order " of an algebraic number field is its ring of integers, and the discriminant of the maximal order is the discriminant the field.
36.
However, even over the ring of integers, the module of rational numbers has an endomorphism ring that is a division ring, specifically the field of rational numbers.
37.
Thus, a number field has class number 1 if and only if its ring of integers is a principal ideal domain ( and thus a unique factorization domain ).
38.
A natural generalisation of the construction above is as follows : let F be a number field with ring of integers O and \ mathrm G an algebraic group over F.
39.
From the point of view of abstract algebra, congruence modulo n is a congruence relation on the ring of integers, and arithmetic modulo n occurs on the corresponding quotient ring.
40.
The greatest common divisor may be defined and exists, more generally, for multivariate polynomials over a field or the ring of integers, and also over a unique factorization domain.
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