where p is an odd prime, has been proved in a variety of ways. In particular the proof in [3, p. 623 ] may be cited. We remark that Estermann [1 ] has recently given a simple proof of (1) that is valid for arbitrary odd p. In the present note we indicate a short proof of (1) that makes use of some familiar results from cyclotomy. Let E = e27riP and let g denote a primitive root (mod p); define the determinant of order p -1 D-= I JEr-I (r, s-= , 1, , p-2). Then it is clear that D is also equal to the determinant A'= Ers'I (r, s = 1, 2, , p 1), where ss' 1 (mod p); this in turn is equal to (-1) (P-8) /2A = (-1) (p-3) /2 1 Ers I (r, s = 1, 2, ..., p1), since it is necessary to make (p 3)/2 interchanges in going from A' to A. In the next place it is known [3, p. 465 ] that f (e r E) = j(P-1)/2p(p-2)/2 1