valuation ring of a field

In this article, K is a field with a nontrivial nonarchimedean absolute valueMathworldPlanetmathPlanetmathPlanetmath (valuation) || and K* its multiplicative groupMathworldPlanetmath of units (nonzero elements).

Proposition 1.

  1. 1.

    A=df{xK|x|1} is a ring, called the valuation ringMathworldPlanetmathPlanetmath of (K,||),

  2. 2.

    𝔪=df{xK|x|<1} is the unique maximal idealMathworldPlanetmath of A, and A*={xK|x|=1},

  3. 3.

    K is the fraction field of A.


For (1), note that 1A, that x,yA|x|1,|y|1|xy|1xyA, and that x,yA|x-y|max(|x|,|-y|)1x-yA.

For (2), it is obvious that 𝔪+𝔪𝔪 and that 𝔪A𝔪 so that 𝔪 is an ideal. Clearly A-𝔪={xK|x|=1} which is obviously A* and the result follows from general considerations regarding units in a local ringMathworldPlanetmath.

Finally, to prove (3), choose some xK with |x|<1 (to do this, choose any z whose valuation is not 1; then either z or z-1 will suffice). Given yK*, there is some n such that |y||x|n<1, so that yxnA and thus


is in the fraction field of A. ∎

We say that the absolute value || is discrete if |K*| is a discrete subgroup of >0. Note that >0(,+) via log, so discrete subgroups are isomorphicPlanetmathPlanetmathPlanetmath to (are a lattice in ), and thus a discrete absolute value is of the form |K*|=α for some α1, and α=1 corresponds to the trivial absolute value.

Proposition 2.

In the notation of the preceding theorem, TFAE:

  1. 1.

    A is principal

  2. 2.

    || is discrete

  3. 3.

    A is NoetherianPlanetmathPlanetmathPlanetmath

If any of these hold, A is a discrete valuation ring (DVR).


(12): If A is principal, then 𝔪=(π) with |π|<1. Since A is a UFD, any element xA-{0} can be written uniquely as x=uπn for uA*,n0, and then |x|=|u||π|n=|π|n. Thus |A-{0}|=|π| and |K*|=|π| so that || is discrete.

(21): If the absolute value is discrete, we may choose πK* with |π|<1 but with the largest possible absolute value strictly less than 1. Then for x𝔪, we have |x|<1, so |x||π| and thus |xπ|1 so that xπA. It follows that xπA=(π), so A is principal.

Clearly principal implies Noetherian, so it suffices to prove that 32: if || is not discrete, then A 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 subgroupMathworldPlanetmathPlanetmath of , we can find a convergent sequence (rn) with rn+1>rn, limrn=1, and a sequence of elements of A with |xn|=rn. Now consider In={xA,|x|rn}. Then


and xn+1In+1\In, so that A is not Noetherian.

The fact that A 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