## Hensel’s lemma and a classification theoremSeptember 2, 2009

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