Unit groups for complex dihedral extensions of degree 10 over \(\mathbb Q\)
H. Darmon exhibited nice properties of the roots of \[ p(X) = X^5 - SX^4 + (S+ T+5)X^3-(S^2 +S- 2T -5)X^2 +(2S+ T+5)X+(S+3) \] \[ =(X-\theta_1)(X-\theta_2)(X -\theta_3)(X-\theta_4)(X- \theta_5) \] which satisfy the (nonlinear) recurrence of order 5: \(\theta_{1-1}\theta_{1+1}=\theta_1+1\). In the paper under review, the author proves that if \(S= -2\) (resp. \(S= -4\)) and \(T > 385000\) (resp. \(T\geq 141590\)), then any set of four roots forms a fundamental system of units of \(F =\mathbb{Q}(\theta_1, \theta_2, \theta_3, \theta_4, \theta_5)\). Here the field \(F\) is a Galois extension of \(\mathbb{Q}\) whose Galois group is dihedral of order 10. This is a very nice result, and the proof is based upon some ideas of \textit{R. Schoof} and \textit{L. Washington} in their paper [Math. Comput. 50, 543-556 (1988; Zbl 0649.12007)] on the unit group of some families of quintic cyclic fields.
scientific article; zbMATH DE number 1770782
scientific article