  Let $E_p(x)$ denote the Artin-Hasse exponential and let $\overline{E}_p(x)$
denote its reduction modulo $p$ in $\mathbb{F}_p[[x]]$. In this article we
study transcendence properties of $\overline{E}_p(x)$ over $\mathbb{F}_p[x]$.
We give two proofs that $\overline{E}_p(x)$ is transcendental, affirmatively
answering a question of Thakur. We also prove algebraic independence results:
i) for $f_1,\dots,f_r \in x\mathbb{F}_p[x]$ satisfying certain linear
independence properties, we show that the $\overline{E}_p(f_1), \dots,
\overline{E}_p(f_r)$ are algebraically independent over $\mathbb{F}_p[x]$ and
ii) we determine the algebraic relations between $\overline{E}_p(cx)$, where $c
\in \mathbb{F}_p^\times$. Our proof studies the higher derivatives of
$\overline{E}_p(x)$ and makes use of iterative differential Galois theory.
