Kihel, Omar Unit groups for complex dihedral extensions of degree 10 over \(\mathbb Q\). (Groupe des unités pour des extensions diédrales complexes de degré 10 sur \(\mathbb Q\).) (French) Zbl 1012.11096 J. Théor. Nombres Bordx. 13, No. 2, 469-482 (2001). 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 R. Schoof and 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. Reviewer: Claude Levesque (Quebec) Cited in 3 Documents MSC: 11R27 Units and factorization 11R32 Galois theory Keywords:dihedral Galois group; fundamental system of units; Galois extension Citations:Zbl 0649.12007 PDFBibTeX XMLCite \textit{O. Kihel}, J. Théor. Nombres Bordx. 13, No. 2, 469--482 (2001; Zbl 1012.11096) Full Text: DOI Numdam EuDML EMIS References: [1] Berwick, W.E.H., Algebraic number fields with two independent units. Proc. London Math. Soc34 (1932), 360-378. · JFM 58.0175.02 [2] Billevi, K.K., Sur les unités d’un corps algébrique de degré 3 ou 4. Mat. Sbornik N. S.40 (1956) (en russe). [3] Brumer, A., On the group of units of an absolutely cyclic number field of prime degree. J. Math. Soc. Japan21 (1969), 357-358. · Zbl 0188.35301 [4] Cusick, T.W., Lower bounds for regulators. 1068, 63-73, Springer, Berlin, 1984. · Zbl 0549.12003 [5] Darmon, H., Une famille de polynômes liée à X0(5). Notes non publiées, 1993. [6] Darmon, H., Note on a polynomial of Emma Lehmer. Math. Comp.56 (1991), 795-800. · Zbl 0732.11056 [7] Edwards, M., Galois Theory. 101, Springer-Verlag, New York, 1984. · Zbl 0532.12001 [8] Gras, M.-N., Special units in real cyclic sextic fields. Math. Comp.48 (1987), 179-182. · Zbl 0617.12006 [9] Ishida, M., Fundamental units of certain algebraic number fields. Abh. Math. Semi.Univ. Hamburg39 (1973), 245-250. · Zbl 0311.12005 [10] Iwasawa, K., A note on the group of units of an algebraic number field. J. Math. Pures Appl.35 (1956), 189-192. · Zbl 0071.26504 [11] Lecacheux, O., Unités d’une famille de corps liés à la courbe X1(25). Ann. Inst. Fourier40 (1990), 237-254. · Zbl 0739.11023 [12] Lehmer, E., Connections between Gaussian periods and cyclic units. Math. Comp.50 (1988), 535-541. · Zbl 0652.12004 [13] Maki, S., The determination of units in real cyclic sextic fields. 797, Springer, Berlin, 1980. · Zbl 0423.12006 [14] Shanks, D., The simplest cubic fields. Math. Comp.28 (1974), 1137-1152. · Zbl 0307.12005 [15] Schoof, R., Washington, L.C., Quintic polynomials and real cyclotomic fields with large class number. Math. Comp.50 (1988), 541-555. · Zbl 0649.12007 [16] Stender, H.-J., Lôsbare Gleichungen axn - byn = c und Grundeinheiten fûr einige algebraische Zahikôrper vom Grade n = 3,4,6. J. Reine Angew. Math.290 (1977), 24-62. · Zbl 0499.12004 [17] Washington, L.C., Introduction to Cyclotomic Fields. 83, Springer-Verlag, New York, 1982. · Zbl 0484.12001 [18] Weiss, E., First Course in algebra and number theory. Academic Press, New York-London, 1971. · Zbl 0241.10001 [19] Zhao, C.L., The fundamental units in absolutely cyclic number fields of degree five. Sci. Sinica Ser. A27 (1984), 27-40. · Zbl 0531.12005 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.