RO(S1)-graded TR-groups of Fp, Z and ℓ

We give an algorithm for calculating the RO(S1)-graded TR-groups of Fp, completing the calculation started by the second author. We also calculate the RO(S1)-graded TR-groups of Z with mod p coefficients and of the Adams summand ℓ of connective complex K-theory with V(1)-coefficients. These calculations are used elsewhere to compute the algebraic K-theory of certain Z-algebras.