A Theorem in Finite Projective Geometry and Some Applications to Number Theory

A point in a finite projective plane PG(2, pn), may be denoted by the symbol (xi, x2, x3), where the coordinates xi, x2, x3 are marks of a Galois field of order pn, GF(pn). The symbol (0, 0, 0) is excluded, and if k is a non-zero mark of the GF(pn), the symbols (xi, Xi, x3) and (kxh kx2, kx3) are to be thought of as the same point. The totality of points whose coordinates satisfy the equation uiXi+u2x2+u3x3 = 0, where ui, w2, u3 are marks of the GF(pn), not all zero, is called a line. The plane then consists of p2n+pn+l = q points and q lines; each line contains pn + \ points, j A finite projective plane, PG(2, pn), defined in this way is Pascalian and Desarguesian ; it exists for every prime p and positive integer , and there is only one such PG(2, pn) for a given p and (VB, p. 247, VY, p. 151).