Transcendence properties of the Artin-Hasse exponential modulo p

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