Previous |  Up |  Next


cyclotomic $\mathbb{Z}_2$-extension; $2$-rank; $2$-class group
Let $d$ be an odd square-free integer, $m\ge 3$ any integer and $L_{m, d}:=\mathbb{Q}(\zeta _{2^m},\sqrt{d})$. In this paper, we shall determine all the fields $L_{m, d}$ having an odd class number. Furthermore, using the cyclotomic $\mathbb{Z}_2$-extensions of some number fields, we compute the rank of the $2$-class group of $L_{m, d}$ whenever the prime divisors of $d$ are congruent to $3$ or $5\pmod 8$.
[1] Azizi, A., Chems-Eddin, M.M., Zekhnini, A.: On the rank of the $2$-class group of some imaginary triquadratic number fields. Rend. Circ. Mat. Palermo, II Ser. (2019), 19 pp., DOI 10.1007/s12215-020-00589-0
[2] Azizi, A., Zekhnini, A., Taous, M.: On the strongly ambiguous classes of some biquadratic number fields. Math. Bohem. 14 (2016), 363–384. DOI 10.21136/MB.2016.0022-14 | MR 3557585
[3] Chems-Eddin, M.M., Müller, K.: $2$-class groups of cyclotomic towers of imaginary biquadratic fields and applications. Accepted for publication in Int. J. Number Theory, arXiv:2002.03602.
[4] Connor, P.E., Hurrelbrink, J.: Class number parity. Series in Pure Mathematics, World Scientific, 1988. MR 0963648
[5] Fukuda, T.: Remarks on $\mathbb{Z}_p$-extensions of number fields. Proc. Japan Acad. Ser. A Math. Sci. 70 (1994), 264–266. DOI 10.3792/pjaa.70.264 | MR 1303577
[6] Gras, G.: Sur les l-classes d’idéaux dans les extensions cycliques relatives de degré premier l. Ann. Inst. Fourier (Grenoble) 23 (1973), 1–48. DOI 10.5802/aif.480 | MR 0360519
[7] Hilbert, D.: Über die Theorie des relativquadratischen Zahlkörpers. Math. Annal. 51 (1898), 1–127. DOI 10.1007/BF01905120
[8] Hubbard, D., Washington, L.C.: Iwasawa Invariants of some non-cyclotomic $\mathbb{Z}$-extensions. arXiv:1703.06550. MR 3778621
[9] Lemmermeyer, F.: Kuroda’s class number formula. Acta Arith. 66 (1994), 245–260. DOI 10.4064/aa-66-3-245-260 | MR 1276992 | Zbl 0807.11052
[10] Li, J., Ouyang, Y., Xu, Y., Zhang, S.: $l$-class groups of fields in Kummer towers. arXiv:1905.04966.
[11] Masley, J.M., Montgomery, H.L.: Cyclotomic fields with unique factorization. J. Reine Angew. Math. 286/287 (1976), 248–256. MR 0429824
[12] McCall, T.M., Parry, C.J., Ranalli, R.R.: Imaginary bicyclic biquadratic fields with cyclic $2$-class group. J. Number Theory 53 (1995), 88–99. DOI 10.1006/jnth.1995.1079 | MR 1344833
[13] Mouhib, A., Movahhedi, A.: Cyclicity of the unramified Iwasawa module. Manuscripta Math. 135 (2011), 91–106. DOI 10.1007/s00229-010-0407-8 | MR 2783388
[14] Washington, L.C.: Introduction to cyclotomic fields. Graduate Texts in Mathematics, vol. 83, Springer-Verlag, New York, second ed., 1997. MR 1421575
Partner of
EuDML logo