Relative norm of the fundamental unit of certain biquadratic fields and parity of the lengths of cycles of reduced ideals (Q1201740)

Using a complex quadratic field \(k\) with class number one as the base field, the author considers a quadratic extension \(K\) of \(k\). He shows that the theory of cycles of reduced ideals developed by H. Amara provides us with an algorithm to test the relative norm \(N_{K/k}({\mathcal N}_ k)\) of the fundamental unit of \(K\). There is a result going back to Lagrange which says that the norm of the fundamental unit of a real quadratic field is \((-1)^ \ell\) where \(\ell\) is the period length of the continued fraction expansion of the principal class. The author shows that this does not generalize to his case.