On the polynomiality of invariant rings for codimension one -modules

Let k be an algebraically closed field of positive characteristic and let be its additive group. Let act linearly on a finite-dimensional vector space V, let k[V] be the ring of polynomial functions on V, and let be the ring of -invariants. In this article, we show that the ring of -invariants is a polynomial ring under the assumption that V is a codimension one -module.