On the quadratic character of quadratic units

AbstractLet p≡1(mod4) be a prime. Let a,b∈Z with p∤a(a2+b2). In the paper we mainly determine (b+a2+b22)p−12(modp) by assuming p=c2+d2 or p=Ax2+2Bxy+Cy2 with AC−B2=a2+b2. As an application we obtain simple criteria for εD to be a quadratic residue (modp), where D>1 is a squarefree integer such that D is a quadratic residue of p, εD is the fundamental unit of the quadratic field Q(D) with negative norm. We also establish the congruences for U(p±1)/2(modp) and obtain a general criterion for p|U(p−1)/4, where {Un} is the Lucas sequence defined by U0=0, U1=1 and Un+1=bUn+k2Un−1 (n⩾1)