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October 30

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Roots of in

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Copied from WP:RD/MA/2020 October 22#Number of roots of polinomial in fields other than C.  --Lambiam 14:58, 30 October 2020 (UTC)[reply]

Experimentally, it appears that the polynomial has roots in iff its degree is odd. In the latter case we always have the root and usually this is the only root. But sometimes there are more: for example, has 3 roots in ( ), has 39 roots in and has 105 roots in I see no obvious pattern, but no doubt someone has studied this.  --Lambiam 08:05, 27 October 2020 (UTC)[reply]

If is an odd prime, the absence of roots of in follows from Fermat's little theorem. For composite odd moduli, their absence remains unexplained.  --Lambiam 17:50, 28 October 2020 (UTC)[reply]

The Chinese remainder theorem covers the case of products of distinct odd primes.--Jasper Deng (talk) 03:53, 29 October 2020 (UTC)[reply]
@JD — Can you give a proof sketch? I don't see the connection straightaway. I also remain curious about the odd distribution of the number of roots if the degree is odd.  --Lambiam 15:05, 30 October 2020 (UTC)[reply]
If n is even then (-1)n-1 = -1, and so Xn-1+1=0 has a solution. The number of solutions is a bit trickier to determine. Note that since the map X→Xa is an endomorphism of the multiplicative group mod n, the number of solutions to Xa=b is equal to either 0 or the order of the kernel of this endomorphism. The n odd case seems more difficult, but the structure of the multiplicative group mod n is well known and this information can be utilized to derive a contradiction if Xn-1=-1 has a solution mod n. There may be a more elementary proof but I didn't see one. I haven't completely checked the details, but I think you can go a bit further: If a is even, n odd, a = 2αb where b is odd, then Xa=-1 has a solution mod n only if each prime dividing n is congruent to 1 mod 2α+1. --RDBury (talk) 22:48, 30 October 2020 (UTC)[reply]
PS. OEISA063994 gives the number of solutions of Xn-1=1 (mod n). If n is even this is the same number of solutions as Xn-1=-1. An easy(er) way to see this is that Xn-1=1 iff (-X)n-1=-1. The first even number for which this value is >1 is 28, then 52, 66, 70, 76, ... --RDBury (talk) 07:49, 31 October 2020 (UTC)[reply]