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In analogy with the Riemann zeta function at positive integers, for eachfinite field F_p^r with fixed characteristic p we consider Carlitz zeta valueszeta_r(n) at positive integers n. Our theorem asserts that among the zetavalues in {zeta_r(1), zeta_r(2), zeta_r(3), ... | r = 1, 2, 3, ...}, all thealgebraic relations are those algebraic relations within each individual family{zeta_r(1), zeta_r(2), zeta_r(3), ...}. These are the algebraic relationscoming from the Euler-Carlitz relations and the Frobenius relations. To provethis, a motivic method for extracting algebraic independence results fromsystems of Frobenius difference equations is developed.

Frobenius difference equations and algebraic independence of zeta values in positive equal characteristic