### Quelques classes caracteristiques en theorie des nombres, by Max Karoubi and Thierry Lambre

Let A be an arbitrary ring. We introduce a Dennis trace map mod n, from
K_1(A;Z/n) to the Hochschild homology group with coefficients
HH_1(A;Z/n). If A is the ring of integers in a number field, explicit
elements of K_1(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.

Max Karoubi and Thierry Lambre <karoubi@math.jussieu.fr,thierry.lambre@math.u-psud.fr>