On 2-class field towers of imaginary quadratic number fields
Journal de Théorie des Nombres de Bordeaux, Tome 6 (1994) no. 2, pp. 261-272.

For a number field k, let k 1 denote its Hilbert 2-class field, and put k 2 =(k 1 ) 1 . We will determine all imaginary quadratic number fields k such that G=Gal(k 2 /k) is abelian or metacyclic, and we will give G in terms of generators and relations.

@article{JTNB_1994__6_2_261_0,
     author = {Lemmermeyer, Franz},
     title = {On $2$-class field towers of imaginary quadratic number fields},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {261--272},
     publisher = {Universit\'e Bordeaux I},
     volume = {6},
     number = {2},
     year = {1994},
     zbl = {0826.11052},
     mrnumber = {1360645},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_1994__6_2_261_0/}
}
Lemmermeyer, Franz. On $2$-class field towers of imaginary quadratic number fields. Journal de Théorie des Nombres de Bordeaux, Tome 6 (1994) no. 2, pp. 261-272. http://www.numdam.org/item/JTNB_1994__6_2_261_0/

[1] E. Benjamin, C. Snyder, Number fields with 2-class groups isomorphic to (2, 2m), Austr. J. Math.

[2] M. Hall, J.K. Senior, The groups of order 2n(n ≤ 6);, Macmillan, New York (1964). | Zbl 0192.11701

[3] H. Hasse, Zahlbericht, Physica Verlag, Würzburg, 1965.

[4] H. Hasse, Über die Klassenzahl abelscher Zahlkörper, Springer Verlag, Heidelberg. | Zbl 0668.12004

[5] K. Iwasawa, A note on the group of units of an algebraic number field, . Math. pures appl. 35 (1956), 189-192. | MR 76803 | Zbl 0071.26504

[6] P. Kaplan, Sur le 2-groupe des classes d'idéaux des corps quadratiques, J. reine angew. Math. 283/284 (1974), 313-363. | EuDML 151737 | MR 404206 | Zbl 0337.12003

[7] G. Karpilovsky, The Schur multiplier, London Math. Soc. monographs (1987), Oxford. | MR 1200015 | Zbl 0619.20001

[8] H. Kisilevsky, Number fields with class number congruent to 4 mod 8 and Hilbert's theorem 94, J. Number Theory 8 (1976), 271-279. | MR 417128 | Zbl 0334.12019

[9] H. Koch, Über den 2-Klassenkörperturm eines quadratischen Zahlkörpers, J. reine angew. Math. 214/215 (1963), 201-206. | EuDML 150617 | MR 164945 | Zbl 0123.03904

[10] F. Lemmermeyer, Die Konstruktion von Klassenkörpern, Diss. Univ. Heidelberg (1994). | Zbl 0956.11515

[11] L. Rédei, H. Reichardt, Die Anzahl der durch 4 teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlkörpers, J. reine angew. Math. 170 (1933), 69-74. | Zbl 0007.39602

[12] A. Scholz, Über die Lösbarkeit der Gleichung t2 - du2 = -4, Math. Z. 39 (1934), 95-111. | JFM 60.0126.03 | MR 1545490 | Zbl 0009.29402

[13] A. Scholz, Abelsche Durchkreuzung, Monatsh. Math. Phys. 48 (1939), 340-352. | JFM 65.0066.02 | MR 623 | Zbl 0023.21101