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A366526 Prime powers (A246655) q such that 2 is a nonzero square in the finite field F_q.

%I A366526 #24 Oct 28 2023 11:35:16
%S A366526 7,9,17,23,25,31,41,47,49,71,73,79,81,89,97,103,113,121,127,137,151,
%T A366526 167,169,191,193,199,223,233,239,241,257,263,271,281,289,311,313,337,
%U A366526 343,353,359,361,367,383,401,409,431,433,439,449,457,463,479,487,503,521,529
%N A366526 Prime powers (A246655) q such that 2 is a nonzero square in the finite field F_q.
%C A366526 Prime powers q that are congruent to 1 or 7 modulo 8.
%C A366526 Odd prime powers q such that 2^((q-1)/2) = 1 in F_q.
%C A366526 Prime powers q such that x^2 - 2 splits into different linear factors in F_q[x].
%C A366526 Contains the powers of primes congruent to 1 or 7 modulo 8 and the even powers of primes congruent to 3 or 5 modulo 8.
%C A366526 Proposition 1: Suppose that q is not a power of 2, gcd(a,q) = 1, then a is a square in F_q if and only if the Jacobi symbol Jacobi(a,q) = 1.
%C A366526 Proof: a is a square if and only if a^((q-1)/2) == 1 (mod p). We have a^((q-1)/2) = (a^((p-1)/2))^((q-1)/(p-1)) == Jacobi(a,p)^((q-1)/(p-1)) (mod p). Write q = p^e, then by definition, we have Jacobi(a,q) = Jacobi(a,p)^e, so it remains to prove that (q-1)/(p-1) - e = Sum^{e-1}_{i=0} (p^i - 1) is always even, which is obvious.
%C A366526 A trivial corollary would be that if q is a square, then every integer a coprime to q is always a square in F_q (since Jacobi(a,q) = 1 in this case). Indeed, since F_q is the unique quadratic extension of F_{sqrt(q)}, every quadratic polynomial with coefficients in F_{sqrt(q)} splits in F_q.
%C A366526 Proposition 2: Suppose that a == 1 (mod 4), gcd(a,q) = 1, then x^2 - x - (a-1)/4 splits into different linear factors in F_q[x] if and only if Jacobi(q,a) = 1 (or Kronecker(a,q) = 1).
%C A366526 Proof: Proposition 1 deals with the case where q is odd. For even q, we have x^2 - x - (a-1)/4 = x^2 + x + 1, which is reducible over F_q[x] if and only if q is an even power of 2.
%H A366526 Jianing Song, <a href="/A366526/b366526.txt">Table of n, a(n) for n = 1..10000</a>
%e A366526 9 is a term since 2 = -1 = (+-i)^2 in F_9 = F_3(i).
%o A366526 (PARI) isA366526(n) = isprimepower(n) && (n%8==1 || n%8==7)
%Y A366526 Supersequence of A001132.
%Y A366526 Prime powers q such that a is a nonzero square in F_q: A365082 (q=-2), A085759 (q=-1), this sequence (q=2), A365313 (q=3).
%K A366526 nonn,easy
%O A366526 1,1
%A A366526 _Jianing Song_, Oct 12 2023