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.