Groupe des unités pour des extensions diédrales complexes de degré 10 sur Q
Journal de théorie des nombres de Bordeaux, Volume 13 (2001) no. 2, p. 469-482

The purpose of this paper is to show that any set of four roots of the quintic polynomials exhibited by H. Darmon forms under certain conditions a fundamental system of units for the corresponding dihedral fields.

Le but de cet article est de montrer qu’un ensemble quelconque de quatre racines des polynômes quintiques p(x) exhibés par H. Darmon forme sous certaines conditions un système fondamental d’unités de la fermeture normale du corps 𝐐(θ)p(θ)=0.

@article{JTNB_2001__13_2_469_0,
     author = {Kihel, Omar},
     title = {Groupe des unit\'es pour des extensions di\'edrales complexes de degr\'e $10$ sur $Q$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux I},
     volume = {13},
     number = {2},
     year = {2001},
     pages = {469-482},
     zbl = {1012.11096},
     mrnumber = {1879669},
     language = {fr},
     url = {http://www.numdam.org/item/JTNB_2001__13_2_469_0}
}
Kihel, Omar. Groupe des unités pour des extensions diédrales complexes de degré $10$ sur $Q$. Journal de théorie des nombres de Bordeaux, Volume 13 (2001) no. 2, pp. 469-482. http://www.numdam.org/item/JTNB_2001__13_2_469_0/

[1] W.E.H. Berwick, Algebraic number fields with two independent units. Proc. London Math. Soc 34 (1932), 360-378. | JFM 58.0175.02 | Zbl 0005.34203

[2] K.K. Billevi, Sur les unités d'un corps algébrique de degré 3 ou 4. Mat. Sbornik N. S. 40 (1956) (en russe).

[3] A. Brumer, On the group of units of an absolutely cyclic number field of prime degree. J. Math. Soc. Japan 21 (1969), 357-358. | MR 244193 | Zbl 0188.35301

[4] T.W. Cusick, Lower bounds for regulators. Lecture Notes in Math. 1068, 63-73, Springer, Berlin, 1984. | MR 756083 | Zbl 0549.12003

[5] H. Darmon, Une famille de polynômes liée à X0(5). Notes non publiées, 1993.

[6] H. Darmon, Note on a polynomial of Emma Lehmer. Math. Comp. 56 (1991), 795-800. | MR 1068821 | Zbl 0732.11056

[7] M. Edwards, Galois Theory. Graduate Texts in Mathematics 101, Springer-Verlag, New York, 1984. | MR 743418 | Zbl 0532.12001

[8] M.-N. Gras, Special units in real cyclic sextic fields. Math. Comp. 48 (1987), 179-182. | MR 866107 | Zbl 0617.12006

[9] M. Ishida, Fundamental units of certain algebraic number fields. Abh. Math. Semi.Univ. Hamburg 39 (1973), 245-250. | MR 335469 | Zbl 0311.12005

[10] K. Iwasawa, A note on the group of units of an algebraic number field. J. Math. Pures Appl. 35 (1956), 189-192. | MR 76803 | Zbl 0071.26504

[11] O. Lecacheux, Unités d'une famille de corps liés à la courbe X1(25). Ann. Inst. Fourier 40 (1990), 237-254. | Numdam | MR 1070827 | Zbl 0739.11023

[12] E. Lehmer, Connections between Gaussian periods and cyclic units. Math. Comp. 50 (1988), 535-541. | MR 929551 | Zbl 0652.12004

[13] S. Maki, The determination of units in real cyclic sextic fields. Lecture Notes in Math. 797, Springer, Berlin, 1980. | MR 584794 | Zbl 0423.12006

[14] D. Shanks, The simplest cubic fields. Math. Comp. 28 (1974), 1137-1152. | MR 352049 | Zbl 0307.12005

[15] R. Schoof, L.C. Washington, Quintic polynomials and real cyclotomic fields with large class number. Math. Comp. 50 (1988), 541-555. | MR 929552 | Zbl 0649.12007

[16] H.-J. Stender, 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. | MR 472765 | Zbl 0499.12004

[17] L.C. Washington, Introduction to Cyclotomic Fields. Graduate Texts in Mathematics 83, Springer-Verlag, New York, 1982. | MR 718674 | Zbl 0484.12001

[18] E. Weiss, First Course in algebra and number theory. Academic Press, New York-London, 1971. | MR 278864 | Zbl 0241.10001

[19] C.L. Zhao, The fundamental units in absolutely cyclic number fields of degree five. Sci. Sinica Ser. A 27 (1984), 27-40. | MR 764154 | Zbl 0531.12005