  Two finite Alexander quandles with the same number of elements are isomorphic
iff their Z[t,t^-1]-submodules Im(1-t) are isomorphic as modules. This yields
specific conditions on when Alexander quandles of the form Z_n[t,t^-1]/(t-a)
where gcd(n,a)=1 (called linear quandles) are isomorphic, as well as specific
conditions on when two linear quandles are dual and which linear quandles are
connected. We apply this result to obtain a procedure for classifying Alexander
quandles of any finite order and as an application we list the numbers of
distinct and connected Alexander quandles with up to fifteen elements.
