  In this paper we give a counterexample to Weitzenb\"{o}ck's Theorem in
positive characteristic. Namely we show that if $ k $ is an algebraically
closed field of characteristic $ p $, there is an action of $ \mathbb{G}_{a} $
on $ \mathbb{A}^{5}_{k} $, induced by a linear representation of $
\mathbb{G}_{a} $, such that the ring of invariants $
k[x_{1},\dots,x_{5}]^{\mathbb{G}_{a}} $ is not finitely generated.
