When is a discrete valuation ring a PID? Must it have an element of valuation 1 or is this not necessary?
2026-04-25 00:36:38.1777077398
For a discrete valuation ring to be a PID, must it have an element of valuation 1?
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Since your valuation is discrete, it's value group is $\mathbb{Z}\alpha$ for some $\alpha\in\mathbb{R}$. Then, the maximal ideal is generated by any element with valuation $\alpha$. It may not be true that every discrete valuation has an element of valuation 1, but for purely cosmetic reasons. Every discrete valuation is equivalent to one which has an element of valuation $1$.