Abstract :
Let F be a totally real cubic field. Then, the group of totally positive units in F acts on R3 in a natural way. This paper gives an elementary geometric proof for the construction of a fundamental domain for this action by Thomas and Vasquez in [13]. Furthermore, an explicit class number formula for a totally imaginary quadratic extension of F is described. This formula is completely effective, since a method for determining the Hasse′s unit index is given. Some numerical examples of class numbers are included.
