Ayadi, Mohammed; Azizi, Abdelmalek; Ismaili, Moulay Chrif Ismaili

The capitulation problem for certain number fields. Miyake, Katsuya (ed.), Class field theory – its centenary and prospect. Proceedings of the 7th MSJ International Research Institute of the Mathematical Society of Japan, Tokyo, Japan, June 3-12, 1998. Tokyo: Mathematical Society of Japan. Adv. Stud. Pure Math. 30, 467-482 (2001)

Let k be an algebraic number field of which the 2-class group Ck,2 is of type (2, 2). H. Kisilevsky classified the capitulation of classes of Ck,2 in the 2-class field of k into three types (Type 1-3). This part deals with the classification for k = Q(\sqrt{d},\sqrt{−1}) with positive integer d. The results give explicit necessary and sufficient conditions for k that Ck,2 is of type (2, 2), and give in that case the above classifications of the capitulations in terms of Galois theory in some class fields of k. The second part explains the first author’s results. Let k be a cubic cyclic field over Q of which the 3-class group Ck,3 is of type (3, 3), and let k(�) denote its absolute genus field. In the case where [k(�) : k] = 3 the capitulation of classes of Ck,3 is classified using the cases whether 3 divide exactly hk(�) or 9 divide exactly hk(�) . Further if [k(�) : k] = 9, then the capitulation of classes of Ck,3 is determined in several cases. The third part explains the third author’s results. Let 􀀀 = Q( \sqrt{3}{n}) be a pure cubic field, k0 = Q(e2i�3 ), and k = 􀀀k0 be its normal closure. Let Sk denote the 3-ideal class group of k and assume Sk is of type (3, 3). Then there are 4 intermediate extensions Ki of k(3)1 /k of degree 3, where k(3)1 denotes the Hilbert 3-class field of k. Let (k/k0)(�) denote the relative genus field of k over k0. The author divides k into three types I, II and III according to (k/k0)(�) = k􀀀1, (k/k0)(�) 6= k􀀀1 with (k/k0)� a proper subfield of k(3)1 , and (k/k0)(�) = k(3)1 , respectively, where 􀀀1 denotes the Hilbert 3-class field of 􀀀. Then for each case the conditions of k and the capitulation of classes of Sk are studied, in particular, possible types of capitulation classes of Sk in Ki’s are given.

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