The aim of this paper is to construct affine embeddings of compact flat manifolds M with cyclic holonomy group of prime order p, called \(Z_ p\)- manifolds, into Euclidean space. First of all, the author shows that every compact flat manifold has the structure of a real affine variety and gives an explicit set of generators of the affine algebra of a \(Z_ p\)-manifold. Then he constructs affine embeddings of all \(Z_ 2\)- manifolds by means of generators of the affine algebra.