Sommes de Dedekind associées à un corps de nombres totalement réel

We extend the construction of Dedekind sums to the case of an arbitrary totally real number field of class number one. Our method is based on the choice of some convenient analogue of the logarithm of Dedekind's η function in this context. We deduce its modular transformation from a Kronecker limit formula established by Asai. It allows us to introduce a generalization of Rademacher's Φ function. We use this function to define the corresponding Dedekind sums and derive their main properties. These sums are not rational numbers but real-analytic functions.