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.