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Let q be an odd prime power. Let n be the multiplicative order of q modulo t, where t is even and t > 4. What are the values of t for which the following claim fails?

Claim: n is the multiplicative order of q modulo t/2.

I came across this as I was computing some minimal polynomials m(x) of alpha^2 where alpha is a root of some irreducible polynomial f(x) over GF(q). I noticed that all the time I was getting deg m(x) = deg f(x) which is interesting. So I did a few examples to see for which t this fails. I tried q = 3, 5, 7, 11. I found that claim fails when q = 1 mod t, or q = 1 mod t/2, butsometimesfails when t = 8, 16, 32... However, sometimes it it fails "mysteriously". For example, when q = 11 claim fails for t = 28. What is the pattern here?

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# Congruence relation

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