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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: , ,

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 {R} be a complete DVR with maximal ideal {\mathfrak{m}} and quotient field {F}. We let {k:=R/\mathfrak{m}}; this is the residue field and is, e.g., the integers mod {p} for the {p}-adic integers (I will discuss this more later).

The main result that we have today is:

Theorem 1 Suppose {k} is of characteristic zero. Then {R \simeq k[[X]]}, the power series ring in one variable, with respect to the usual discrete valuation on {k[[X]]}. (more…)