A Counterexample to Weitzenb\"ock's Theorem in Characteristic $p$

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.