Some theory and exercises on DVRs.

There are several equivalent definitions of DVRs
([Atiyah and Macdonald] list 7). I propose to
start from the very direct definition:

A DVR is a Noetherian local integral domain A
whose maximal ideal is principal m = (z).

Notice that
   f not in m  ==>  f is a unit of A
is included in the definition of local.

Ex 1. Every nonzero  f in A is  z^n*f0  for
some n >= 0 and f0 not in m.

Ex 2. Set K = Frac A. Then every f in K^x
is of the form  f = z^n*f0  for some  n in ZZ
and  f0  a unit of  A.

Ex 3. The function v : K^x -> ZZ defined
by  v(f) = the n in the formula of Ex. 2
is a valuation:

(1) it is a multiplicative homomorphism,
that is v(f1*f2) = v(f1) + v(f2).

(2) v(f1 + f2) >= min { v(f1), v(f2) }

In fact, it is usually equal to the min, and
is  > min only when  v(f1) = v(f2) and the leading
terms cancel.

Ex 4. Conversely, if K is a field with a
surjective valuation v: K^x -> ZZ, then
the subset
 A = { f | f in K and either f = 0 or v(f) >= 0 }
is a DVR. The maximal ideal is { f | v(f) > 0 }.

DVR => UFD => normal ring (integrally closed in
its field of fractions).

The converse is a theorem:

If A is a Noetherian local integral domain
with only primes  0  and m,  and A is normal,
then A is a DVR.

The proof is nontrivial, and I will cover it later.
The condition on a local integral domain that 0 in m
are the only prime ideals is equivalent to Krull
dimension = 1.

Thus we have gone from the most direct definition
to the most abstract: DVR if and only if
local Noetherian integral domain of Krull dim 1.