valuation ring of a field
In this article, is a field with a nontrivial nonarchimedean absolute value (valuation) and its multiplicative group of units (nonzero elements).
Proposition 1.
-
1.
is a ring, called the valuation ring of ,
-
2.
is the unique maximal ideal of , and ,
-
3.
is the fraction field of .
Proof.
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.
Proposition 2.
In the notation of the preceding theorem, TFAE:
-
1.
is principal
-
2.
is discrete
-
3.
is Noetherian
If any of these hold, is a discrete valuation ring (DVR).
Proof.
(): 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 |
Canonical name | ValuationRingOfAField |
Date of creation | 2013-03-22 19:03:25 |
Last modified on | 2013-03-22 19:03:25 |
Owner | rm50 (10146) |
Last modified by | rm50 (10146) |
Numerical id | 4 |
Author | rm50 (10146) |
Entry type | Theorem |
Classification | msc 11R99 |
Classification | msc 12J20 |
Classification | msc 13A18 |
Classification | msc 13F30 |
Related topic | HenselianField |
Related topic | RingOfExponent |
Defines | valuation ring |
Defines | discrete valuation |