Quartic residues and binary quadratic forms

Let p≡1(mod4) be a prime, m∈Z and p∤m. In this paper we obtain a general criterion for m to be a quartic residue (modp) in terms of appropriate binary quadratic forms. Let d>1 be a squarefree integer such that (dp)=1, where (dp) is the Legendre symbol, and let εd be the fundamental unit of the quadratic field Q(d). Since 1942 many mathematicians tried to characterize those primes p so that εd is a quadratic or quartic residue (modp). In this paper we will completely solve these open problems by determining the value of (u+vd)(p-(-1p))/2(modp), where p is an odd prime, u,v,d∈Z,v≠0,gcd(u,v)=1 and (-dp)=1. As an application we also obtain a general criterion for p∣u(p-(-1p))/4(a,b), where {un(a,b)} is the Lucas sequence defined by u0=0,u1=1 and un+1=bun-aun-1(n⩾1).