Finite Vector Spaces and Certain Lattices

Abstract

The Galois number $G_n(q)$ 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 $G_n(q)$ is equal to the number $L_n(q)$ 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 ${\bf Z}^n$ and having the property that given any point $P$ in the lattice, all points of ${\bf Z}^n$ which are congruent to $P$ mod $q$ are also in the lattice. For each $n$, we prove that $L_n(q)$ is a multiplicative function of $q$.