Transcendence properties of the Artin-Hasse exponential modulo p

Let Eₚ (x) denote the Artin-Hasse exponential and let Eₚ (x) denote its reduction modulo p in Fₚ[x]. In this article we study transcendence properties of Eₚ (x) over Fₚx. We give two proofs that Eₚ (x) is transcendental, affirmatively answering a question of Thakur. We also prove algebraic independence results: i) for f₁, , fᵣ xFₚx satisfying certain linear independence properties, we show that the Eₚ (f₁), , Eₚ (fᵣ) are algebraically independent over Fₚx and ii) we determine the algebraic relations between Eₚ (cx), where c Fₚ^. Our proof studies the higher derivatives of Eₚ (x) and makes use of iterative differential Galois theory.