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Keywords:
$2$-class group; imaginary biquadratic number field; capitulation; Hilbert $2$-class field
Summary:
Let $\Bbbk =\mathbb {Q} \bigl (\sqrt 2, \sqrt d \bigr )$ be an imaginary bicyclic biquadratic number field, where $d$ is an odd negative square-free integer and $\Bbbk _2^{(2)}$ its second Hilbert $2$-class field. Denote by $G={\rm Gal}(\Bbbk _2^{(2)}/\Bbbk )$ the Galois group of $\Bbbk _2^{(2)}/\Bbbk $. The purpose of this note is to investigate the Hilbert $2$-class field tower of $\Bbbk $ and then deduce the structure of $G$.
References:
[1] Azizi, A.: Unités de certains corps de nombres imaginaires et abéliens sur $\mathbb Q$. Ann. Sci. Math. Qué. 23 (1999), 15-21 French. MR 1721726 | Zbl 1041.11072
[2] Azizi, A.: Sur les unités de certains corps de nombres de degré 8 sur $\mathbb Q$. Ann. Sci. Math. Qué. 29 (2005), 111-129 French. MR 2309703 | Zbl 1188.11056
[3] Azizi, A., Benhamza, I.: Sur la capitulation des 2-classes d'idéaux de $\mathbb Q(\sqrt d, \sqrt{-2})$. Ann. Sci. Math. Qué. 29 (2005), 1-20 French. MR 2296826 | Zbl 1217.11097
[4] Azizi, A., Chems-Eddin, M. M., Zekhnini, A.: On the rank of the 2-class group of some imaginary triquadratic number fields. Available at https://arxiv.org/abs/1905.01225 (2019), 21 pages.
[5] Azizi, A., Mouhib, A.: Capitulation des 2-classes d'idéaux de $\mathbb Q(\sqrt{2}, \sqrt{d})$ où $d$ est un entier naturel sans facteurs carrés. Acta Arith. 109 (2003), 27-63 French. DOI 10.4064/aa109-1-2 | MR 1980850 | Zbl 1077.11078
[6] Azizi, A., Talbi, M.: Capitulation des 2-classes d'idéaux de certains corps biquadratiques cycliques. Acta Arith. 127 (2007), 231-248 French. DOI 10.4064/aa127-3-3 | MR 2310345 | Zbl 1169.11049
[7] Azizi, A., Taous, M.: Capitulation des 2-classes d'idéaux de $k=\mathbb Q(\sqrt{2p},i)$. Acta Arith. 131 (2008), 103-123 French. DOI 10.4064/aa131-2-1 | MR 2388046 | Zbl 1139.11048
[8] Conner, P. E., Hurrelbrink, J.: Class Number Parity. Series in Pure Mathematics 8. World Scientific, Singapore (1988). DOI 10.1142/0663 | MR 0963648 | Zbl 0743.11061
[9] Gorenstein, D.: Finite Groups. Harper's Series in Modern Mathematics. Harper and Row, New York (1968). MR 0231903 | Zbl 0185.05701
[10] Gras, G.: Sur les $\ell$-classes d'idéaux dans les extensions cycliques rélatives de degré premier $\ell$. I. Ann. Inst. Fourier 23 (1973), 1-48 French. DOI 10.5802/aif.471 | MR 0360519 | Zbl 0276.12013
[11] Heider, F.-P., Schmithals, B.: Zur Kapitulation der Idealklassen in unverzweigten primzyklischen Erweiterungen. J. Reine Angew. Math. 336 (1982), 1-25 German. DOI 10.1515/crll.1982.336.1 | MR 0671319 | Zbl 0505.12016
[12] Ishida, M.: The Genus Fields of Algebraic Number Fields. Lecture Notes in Mathematics 555. Springer, Berlin (1976). DOI 10.1007/BFb0100829 | MR 0435028 | Zbl 0353.12001
[13] Kaplan, P.: Divisibilité par 8 du nombre des classes des corps quadratiques dont le 2-groupe des classes est cyclique, et réciprocité biquadratique. J. Math. Soc. Japan 25 (1973), 596-608 French. DOI 10.2969/jmsj/02540596 | MR 0323757 | Zbl 0276.12006
[14] Kaplan, P.: Sur le 2-groupe des classes d'idéaux des corps quadratiques. J. Reine. Angew. Math. 283-284 (1976), 313-363 French. DOI 10.1515/crll.1976.283-284.313 | MR 0404206 | Zbl 0337.12003
[15] Kisilevsky, H.: Number fields with class number congruent to $4 \pmod 8$ and Hilbert's Theorem 94. J. Number Theory 8 (1976), 271-279. DOI 10.1016/0022-314X(76)90004-4 | MR 0417128 | Zbl 0334.12019
[16] Kučera, R.: On the parity of the class number of a biquadratic field. J. Number Theory 52 (1995), 43-52. DOI 10.1006/jnth.1995.1054 | MR 1331764 | Zbl 0852.11065
[17] 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
[18] 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 | Zbl 0831.11059
[19] Taussky, O.: A remark concerning Hilbert's Theorem 94. J. Reine Angew. Math. 239-240 (1969), 435-438. DOI 10.1515/crll.1969.239-240.435 | MR 0279070 | Zbl 0186.09002
[20] Wada, H.: On the class number and the unit group of certain algebraic number fields. J. Fac. Sci., Univ. Tokyo, Sect. I 13 (1966), 201-209 \99999MR99999 0214565 \goodbreak. MR 0214565 | Zbl 0158.30103
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