Quelques classes caracteristiques en theorie des nombres
Max Karoubi, Thierry Lambre
Abstract
Let A be an arbitrary ring. We introduce a Dennis trace map mod n, from K1(A;Z/n) to the Hochschild homology group with coefficients HH1(A;Z/n). If A is the ring of integers in a number field, explicit elements of K1(A,Z/n) are constructed and the values of their Dennis trace mod n are computed. If F is a quadratic field, we obtain this way non trivial elements of the ideal class group of A. If F is a cyclotomic field, this trace is closely related to Kummer logarithmic derivatives; this trace leads to an unexpected relationship between the first case of Fermat last theorem, K-theory and the number of roots of Mirimanoff polynomials.
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