Algebraic Independence of Carlitz Zeta Values with Varying Constant Fields

As an analogue to special values at positive integers of the Riemann zeta function, for each constant field Fpr with fixed characteristic p we consider Carlitz zeta values ζr(n) at positive integers n. The main theorem of this paper asserts that among the families of Carlitz zeta values ∪∞ r=1 {ζr(1), ζr(2), ζr(3), . . . }, all the algebraic relations are those algebraic relations among each individual family {ζr(1), ζr(2), ζr(3), . . . }, r = 1, 2, 3, . . . .