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Hensel’s lemma and a classification theorem
*September 2, 2009*

*Posted by Akhil Mathew in algebra, algebraic number theory, commutative algebra, number theory.*

Tags: characteristic zero, discrete valuation rings, Hensel's lemma

3 comments

Tags: characteristic zero, discrete valuation rings, Hensel's lemma

3 comments

So, I’ll discuss the proof of a classification theorem that DVRs are often power series rings, using Hensel’s lemma.

**Systems of representatives **

Let be a complete DVR with maximal ideal and quotient field . We let ; this is the **residue field** and is, e.g., the integers mod for the -adic integers (I will discuss this more later).

The main result that we have today is:

Theorem 1Suppose is of characteristic zero. Then , the power series ring in one variable, with respect to the usual discrete valuation on . (more…)