We build on work of Anglès–Pellarin concerning evaluations of the Anderson–Thakur function and its hyperderivatives at roots of unity. Let 𝔭 be a monic irreducible polynomial in A : = 𝔽 q [ θ ] , the ring of polynomials in the indeterminate θ over the finite field 𝔽 q , let ζ be a root of 𝔭 in an algebraic closure of 𝔽 q , and let K : = 𝔽 q ( θ ) . For each positive integer n , let λ n be a generator of the A -module of Carlitz 𝔭 n -torsion. We give a basis for the ring of integers A [ ζ , λ n ] ⊂ K ( ζ , λ n ) over A [ ζ ] ⊂ K ( ζ ) which consists of monomials in the hyperderivatives of the Anderson–Thakur function ω evaluated at the roots of 𝔭 , and which, after suitable ordering, provides an upper triangular, block diagonal representation of the action of Galois. For each n ≥ 2 , we also give an explicit integral element whose Galois orbit provides a field normal basis for the extension K ( ζ , λ n ) / K ( ζ , λ 1 ) .