this post was submitted on 20 Jun 2023
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The square root of -1 is representable in 5-adic integers, but not 2 or 3-adic. Which other p-adic number systems is it integer in? (feddit.uk)
submitted 2 years ago by [email protected] to c/math
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Is there a pattern in p?

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[–] Artisian 9 points 2 years ago* (last edited 2 years ago)

Indeed! 2, as always, is a distraction. As the only even prime, it's the oddest of them all.

The important thing is that -1 is a square modulo p. Once it is you can use the (relatively fun and easy to do) Hensel's Lemma to compute exactly what the sqrt of -1 is in that particular p-adic ring.

So for which primes is -1 a square? A classic result of number theory states that these are the primes p such that p = 1 modulo 4. The primes like 5, 13, and 29 will have square roots of -1 in their p-adic rings, while 7, 11, and 31 will not.

Lots of rewarding math to explore around this question.