On generalised legendre matrices involving roots of unity over finite fields

Motivated by the work initiated by Chapman ['Determinants of Legendre symbol matrices', Acta Arith. 115(2004), 231-244], we investigate some arithmetic properties of the generalized Legendre matrices over finite fields. For example, letting a(1),& ctdot;,a((q-1)/2) be all non-zero squares in the finite field F-q which contains q elements with 2q, we give the explicit value of D(q-1)/2=det[(a(i)+a(j))((q-3)/2)](1 <= i,j <=(q-1)/2). In particular, if q=p is a prime greater than 3, then ((detD(p-1)/2)/(p))={(1)if ((-1)(h(-p)+1)/2 )if p equivalent to 1(mod4), if p equivalent to 3(mod4)and p>3, where ((& sdot;)/(p)) is the Legendre symbol and h(-p) is the class number of Q(root-p).