RO(S¹)-graded TR-groups of F_p, Z and ell

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