On 2-class field towers of imaginary quadratic number fields
Journal de Théorie des Nombres de Bordeaux, Volume 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 = {Franz Lemmermeyer},
     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},
     doi = {10.5802/jtnb.114},
     zbl = {0826.11052},
     mrnumber = {1360645},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.114/}
}
TY  - JOUR
TI  - On $2$-class field towers of imaginary quadratic number fields
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 1994
DA  - 1994///
SP  - 261
EP  - 272
VL  - 6
IS  - 2
PB  - Université Bordeaux I
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.114/
UR  - https://zbmath.org/?q=an%3A0826.11052
UR  - https://www.ams.org/mathscinet-getitem?mr=1360645
UR  - https://doi.org/10.5802/jtnb.114
DO  - 10.5802/jtnb.114
LA  - en
ID  - JTNB_1994__6_2_261_0
ER  - 
%0 Journal Article
%T On $2$-class field towers of imaginary quadratic number fields
%J Journal de Théorie des Nombres de Bordeaux
%D 1994
%P 261-272
%V 6
%N 2
%I Université Bordeaux I
%U https://doi.org/10.5802/jtnb.114
%R 10.5802/jtnb.114
%G en
%F JTNB_1994__6_2_261_0
Franz Lemmermeyer. On $2$-class field towers of imaginary quadratic number fields. Journal de Théorie des Nombres de Bordeaux, Volume 6 (1994) no. 2, pp. 261-272. doi : 10.5802/jtnb.114. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.114/

[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

Cited by Sources: