I read the Wikipedia article and there are a bunch of equivalent definitions that I don’t understand. I understand that it’s a ring with a way to measure the “size” of elements. But the integers isn’t a DVR despite the obvious way to measure the size of integers. Also, they show up in both number theory and the study of algebraic curves, which seems like two completely different concepts.
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Googling, I found your question has been asked before: see https://math.stackexchange.com/questions/4480478/intuition-behind-discrete-valuation-rings.
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