Finite vector spaces and certain lattices

Abstract

The Galois number Gn(q) is defined to be the number of subspaces of the n-dimensional vector space over the finite field GF(q). When q is prime, we prove that Gn(q) is equal to the number Ln(q) of n-dimensional mod q lattices, which are defined to be lattices (that is, discrete additive subgroups of n-space) contained in the integer lattice Z n and having the property that given any point P in the lattice, all points of Zn which are congruent to P mod q are also in the lattice. For each n, we prove that Ln(q) is a multiplicative function of q.