Maximal ideal?
Using a [[valuation.md), \$v(`\pi`{=tex} | valuation.html]], \(v(/pi) = 1\).
uniformizer
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2021-09-14
uniformizer : for \(R\) a DVR, a generator \(\pi\) for the unique maximal ideal, so \(R^{\times}\left\langle{\pi}\right\rangle = R\) and \(x\in R \implies x = u\pi^k\)
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2021-03-26
What is a uniformizer?
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valuation
Any element \(\pi\) for which \(v(\pi) = 1\) is a uniformizer.
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Hensel's Lemma
- Slogan: If a polynomial \(p(x)\) has a simple root \(r\) modulo a prime \(p\), then \(r\) corresponds to a unique root of \(p(x)\) modulo any \(p^n\) gotten by iteratively “lifting” solutions. - - Setup: let \(K\in \mathsf{Field}\) be Complete ring wrt a normalized discrete valuation where \({\mathcal{O}}_K\) is the ring of integers of \(K\) with a uniformizer \(\pi\) and let \(\kappa(k]] \coloneqq{\mathcal{O}}_K/ \left\langle{\pi}\right\rangle\) be the residue field. -
Commutative Algebra Problems
Show that if \(R\) is a Noetherian local integral domain whose maximal ideal is principal, then \(R\) is a PID and thus a DVR. > Hint: look for a uniformizer.