Covering finite fields with cosets of subspaces

AbstractIf V is a vector space over a finite field F, the minimum number of cosets of k-dimensional subspaces of V required to cover the nonzero points of V is established. This is done by first regarding V as a field extension of F and then associating with each coset L of a subspace of V a polynomial whose roots are the points of L. A covering with cosets is then equivalent to a product of such polynomials having the minimal polynomial satisfied by all nonzero points of V as a factor