Unit groups of some multiquadratic number fields and 2-class groups. (English) Zbl 1513.11178

Summary: Let \(p \equiv -q \equiv 5 \pmod 8\) be two prime integers. In this paper, we investigate the unit groups of the fields \(L_1 = Q(\sqrt{2},\sqrt{p},\sqrt{q},\sqrt{-1})\) and \(L_1^+ = Q(\sqrt{2},\sqrt{p},\sqrt{q})\). Furthermore, we give the second 2-class groups of the subextensions of \(L_1\) as well as the 2-class groups of the fields \(L_n = Q(\sqrt{p},\sqrt{q},\zeta_{2^{n+2}})\) and their maximal real subfields.


11R04 Algebraic numbers; rings of algebraic integers
11R27 Units and factorization
11R29 Class numbers, class groups, discriminants
11R37 Class field theory
Full Text: DOI arXiv


[1] Azizi, A., Sur le \(2\)-groupe de classes d’idéaux de \({\mathbb{Q}}(\sqrt{d}, i)\), Rend. Circ. Mat. Palermo, 2, 48, 71-92 (1999) · Zbl 0920.11076
[2] Azizi, A., Sur la capitulation des \(2\)-classes d’idéaux de \(k ={\mathbb{Q}}(\sqrt{2pq}, i)\), où \(p\equiv -q\equiv 1~({\rm mod} \; 4)\), Acta. Arith., 94, 383-399 (2000) · Zbl 0953.11033
[3] Azizi, A., Unités de certains corps de nombres imaginaires et abéliens sur \({\mathbb{Q}} \), Ann. Sci. Math. Québec, 23, 15-21 (1999) · Zbl 1041.11072
[4] Azizi, A., Benhamza, Sur la capitulation des \(2\)-classes d’idéaux de \({\mathbb{Q}}(\sqrt{d}, \sqrt{-2})\), Ann. Sci. Math. Qué., 29, 1-20 (2005) · Zbl 1217.11097
[5] A. Azizi, M.M. Chems-eddin, A. Zekhnini, On the rank of the \(2\)-class group of some imaginary triquadratic number fields. Ren. Circ. Mat. Palermo, II. Ser (2021). doi:10.1007/s12215-020-00589-0 · Zbl 1480.11137
[6] Azizi, A.; Talbi, M., Capitulation des \(2\)-classes d’idéaux de certains corps biquadratiques cycliques, Acta Arith., 127, 231-248 (2007) · Zbl 1169.11049
[7] Azizi, A.; Zekhnini, A.; Taous, M., On the strongly ambiguous classes of some biquadratic number fields, Math. Bohem., 141, 363-384 (2016) · Zbl 1413.11120
[8] M.M. Chems-Eddin, K. Müller, \(2\)-class groups of cyclotomic towers of imaginary biquadratic fields and applications. Int. J. Num. Theory (2021). doi:10.1142/S1793042121500627 · Zbl 1482.11148
[9] Chems-Eddin, MM; Zekhnini, A.; Azizi, A., Units and \(2\)-class field towers of some multiquadratic number fields, Turk J. Math., 44, 1466-1483 (2020) · Zbl 1455.11140
[10] Conner, PE; Hurrelbrink, J., Class Number Parity (1988), Singapore: World Scientific, Singapore · Zbl 0743.11061
[11] Fukuda, T., Remarks on \({\mathbb{Z}}_p\)-extensions of number fields, Proc. Jpn. Acad. Ser. A Math. Sci., 70, 264-266 (1994) · Zbl 0823.11064
[12] Kaplan, P., Sur le \(2\)-groupe des classes d’idéaux des corps quadratiques, J. Reine. Angew. Math., 283, 284, 313-363 (1976) · Zbl 0337.12003
[13] Kisilevsky, H., Number fields with class number congruent to \(4~({\rm mod} \; 8)\) and Hilbert’s theorem \(94\), J. Number Theory, 8, 271-279 (1976) · Zbl 0334.12019
[14] Lemmermeyer, F., Kuroda’s class number formula, Acta Arith., 66, 245-260 (1994) · Zbl 0807.11052
[15] Wada, H., On the class number and the unit group of certain algebraic number fields, J. Fac. Sci. Univ. Tokyo, 13, 201-209 (1966) · Zbl 0158.30103
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.