valuation ring of a field
For (1), note that , that , and that .
For (2), it is obvious that and that so that is an ideal. Clearly which is obviously and the result follows from general considerations regarding units in a local ring.
Finally, to prove (3), choose some with (to do this, choose any whose valuation is not ; then either or will suffice). Given , there is some such that , so that and thus
is in the fraction field of . ∎
We say that the absolute value is discrete if is a discrete subgroup of . Note that via , so discrete subgroups are isomorphic to (are a lattice in ), and thus a discrete absolute value is of the form for some , and corresponds to the trivial absolute value.
(): If is principal, then with . Since is a UFD, any element can be written uniquely as for , and then . Thus and so that is discrete.
(): If the absolute value is discrete, we may choose with but with the largest possible absolute value strictly less than . Then for , we have , so and thus so that . It follows that , so is principal.
Clearly principal implies Noetherian, so it suffices to prove that : if is not discrete, then is not Noetherian. But if the absolute value is not discrete, we can choose a convergent sequence of absolute values and, using the fact that the valuations form an additive subgroup of , we can find a convergent sequence with , , and a sequence of elements of with . Now consider . Then
and , so that is not Noetherian.
The fact that is a DVR follows trivially if any of these conditions holds. ∎
|Title||valuation ring of a field|
|Date of creation||2013-03-22 19:03:25|
|Last modified on||2013-03-22 19:03:25|
|Last modified by||rm50 (10146)|