1. The problem statement, all variables and given/known data Show that the number of distinct solutions of a system of linear equations (in any number of equations, and unknowns) over the field Zp is either 0, or a power of p. 3. The attempt at a solution First off, I was wondering whether there is any difference between "cardinality" of a vector space and "dimension". Aren't both just the size of the basis? (the cardinality of V = dim(V)??) This is just because my prof keeps switching between both and confusing the rest of us. for the question, suppose the system is m equations in n unknowns. The case of 0 is trivial, so if we take the subset W of Zpn of solutions over Zp, W is a vector space, and dim(W) = n. I'm not sure how to put this technically, but for each unknown more than the number of equations (say we parametrize them) we have p possible choices, and thus pt solutions. There's something missing, but am I on the right track?