Farey section in k(i) and k(ρ)

An important tool in the study of the approximation to real numbers by rational numbers is pro- vided by the theory of Farey sections. We develop in this paper an analogous theory for Farey fractions whose numerators and denominators are integers in the quadratic field k(i), or in the quadratic field k(p), where p1 2+p+ 1 =0. This leads us to results on approximation to a complex number by numbers of k(i) or k{p). In particular, we obtain new proofs of two theorems of Minkowski (theorems XIV and XV), which it is hoped are more transparent than those given by Minkowski himself and by Hlawka.