Unit groups of some multiquadratic number fields of degree 16. (English) Zbl 1514.11070

Let \(p\) and \(q\) be two primes, and let \(\ell\) be an odd positive square-free integer distinct from \(1,p,q,pq\). In this paper the authors compute the unit group of some number fields of the form \({\mathbb Q}(\sqrt{2},\sqrt{p},\sqrt{q},\sqrt{-\ell})\). The answer is split into \(7\) cases which depend of the residues of \(p,q\) modulo \(8\) and on whether \(p\) is a quadratic residue modulo \(q\) or not.


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


[1] 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
[2] Chems-Eddin, M.M.: Unit groups of some multiquadratic number fields and \(2\)-class groups, Accepted for publication in Period. Math. Hung. arXiv:2004.08899 · Zbl 1513.11178
[3] Chems-Eddin, M.M.: The 2-Iwasawa module of some imaginary triquadratic fields. arXiv:2007.05953 · Zbl 1500.11081
[4] Chems-Eddin, M.M., Azizi, A., Zekhnini, A.: Unit groups and Iwasawa Lambda invariants of some multiquadratic number fields. Accepted for publication in Bol. Soc. Mat. Mex · Zbl 1468.11223
[5] 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
[6] Chems-Eddin, M.M., Zekhnini, A., Azizi, A.: On theHilbert 2-class field towers of some cyclotomic \({\mathbb{Z}}_2\)-extensions, arXiv:2005.06646 · Zbl 1455.11140
[7] Hirabayashi, M., Unit indices of some imaginary composite quadratic fields II, Pac. J. Math., 173, 93-104 (1996) · Zbl 0849.11084
[8] Hirabayashi, M.; Yoshino, K., Remarks on unit indices of imaginary abelian number fields II, Manuscr. Math., 64, 235-251 (1989) · Zbl 0703.11054
[9] Hirabayashi, M.; Yoshino, K., Unit indices of imaginary abelian number fields of type (2, 2, 2), J. Number Theory, 34, 346-361 (1990) · Zbl 0705.11065
[10] Hirabayashi, M., Unit indices of some imaginary composite quadratic fields, Pac. J. Math., 164, 87-104 (1994) · Zbl 0796.11049
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