I was initially planning on doing a post on Hensel’s lemma. Actually, I think I’ll leave that for later, after I’ve covered some more number theory (which may motivate it better).
So the goal for the next several posts is to cover some algebraic number theory, eventually leading into class field theory. At least in the near future, I intend to keep everything purely local. Thus, the appropriate place to start is to discuss discrete valuation rings rather than Dedekind domains.
Actually, it is perhaps more logical to introduce discrete valuations as a special case of absolute values, which in turn generalize the standard absolute value on .
Definition 1 Let be a field. An absolute value on is a function , satisfying the following conditions:
- for with , and .
- for all .
- (Triangle inequality.)
So for instance , i.e. .
The standard example is of course the normal absolute value on or , but here is another:
Example 1 For prime, let be the -adic absolute value on defined as follows: if with , and are the highest powers of dividing respectively, then
It can be checked directly from the definition that the -adic absolute value is indeed an absolute value, though there are some strange properties: a number has a small -adic absolute value precisely when it is divisible by a high power of .
Moreover, by elementary number theory, it satisfies the nonarchimedean property:
Definition 2 An absolute value on a field is nonarchimedean if .
This is a key property of the -adic absolute value, and what distinguishes it fundamentally from the regular absolute value restricted to . In general, there is an easy way to check for this:
Proposition 3 The absolute value on is non-archimedean if and only if there is a with for all (by abuse of notation, we regard as an element of as well, even when is of nonzero characteristic and the map is not injective). In this case, we can even take .
One way is straightforward: if is non-archimedean, then , , , and so on inductively.
The other way is slightly more subtle. Suppose for . Then fix . We have:
Now by the hypothesis,
Taking -th roots and letting gives the result.
This also shows that the -adic absolute value is nonarchimedean, since it it automatically on the integers.
Corollary 4 If has nonzero characteristic, then any absolute value on is non-archimedean.
Indeed, if is of characteristic , take .
Discrete Valuation Rings
The absolute values we are primarily interested in are
Definition 5 A discrete valuation is an absolute value on a field such that is a cyclic group.
In other words, there is such that, for each , we can write , where is an integer depending on . We assume without loss of generality that , in which case is the order function (sometimes itself called a valuation). Furthermore we assume surjective by choosing as a generator of the cyclic group .
Now, if is any nonarchimedean absolute value on a field , define the ring of integers as
(This is a ring.) Note that is a non-unit if and only if , so the sum of two non-units is a non-unit and is a local ring with maximal ideal
When is a discrete valuation, we call the ring of integers so obtained a discrete valuation ring (DVR). The first thing to notice is:
Proposition 6 A discrete valuation ring is a principal ideal domain. Conversely, a local principal ideal domain is a discrete valuation ring.
Indeed, if is a DVR and if is an ideal, let be an element of maximal order ; then , since consists of the elements of of nonnegative order.
Conversely, if is a local PID, then let generate the maximal ideal ; then since the Krull intersection theorem (Cor. 6 here) implies
we can write each nonzero , say as for , i.e. a unit. This is unique and we can define a discrete valuation by for as above. This extends to the quotient field and makes a DVR.
A much more interesting (and nontrivial) result is the following:
Theorem 7 If is Noetherian, integrally closed (in its quotient field), and has a unique nonzero prime ideal , then is a DVR.
This is equivalent to the fact that Dedekind domains have unique factorization, but I’m only going to be able to get to it in the next post.