We prove here a tropical version of the well-known Whitney embedding theorem (1944) stating that a smooth m-dimensional compact manifold can be embedded into R^2m+1.

The tropical version of this theorem states that a tropical torsion module with m generators can always be embedded into the free tropical module R^p, where p=2 for m=2, and 3≤ p ≤ m(m-1) otherwise, is the number of rows supporting the torsion, when the generators are given by the (independent) columns of a matrix of size n x m.

As a corollary, we get that tropical m-dimensional torsion modules are classified by a (m-1) (m(m-1)-1) parameter family.