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by t-ob
3405 days ago
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> I'm sure you can derive all sorts of weird metrics so that various weird identities are true On the rational numbers, at least, the p-adic metrics are more or less your whole lot, according to Ostrowski's Theorem [1]. There is a kind of cognitive hurdle everyone who studies these numbers has to clear, in that things that should be "large" turn out to be very small indeed, when viewed under a p-adic lens. I think it's more instructive to build up the ring of p-adic integers first [2, chapter 2], and construct the p-adic numbers from there. I can assure you they are very useful, though! A general theme in number theory is to take a "global" problem, defined over the integers, and to translate it into infinitely many "local" ones (over the p-adics, for each prime p). These are sometimes easier to solve and, if you're lucky, offer insight into the global solution you're looking for. [1]: https://en.wikipedia.org/wiki/Ostrowski's_theorem [2]: http://www.springer.com/gb/book/9780387900407 |
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