## Imaginary quadratic fields $$k$$ with cyclic $$\text{Cl}_2(k^1)$$.(English)Zbl 0919.11074

Let $$k$$ be an imaginary quadratic number field, and $$Cl_2(k)$$ denote the 2-class group of $$k$$. Let $$k^1$$ denote the Hilbert 2-class field of $$k$$, and $$k^2=(k^1)^1$$. If $$G =\text{Gal}(k^2/k)$$, then the commutator subgroup $$G'= \text{Gal}(k^2/k^1)\simeq Cl_2(k^1)$$, and $$G^{ab}= G/G'\simeq \text{Gal} (k^1/k) \simeq Cl_2(k)$$. Moreover it is known that if $$G/G'\simeq (2,4)$$ then rank $$Cl_2(k^1)\leq 3$$, and further if rank $$Cl_2(k^1)<3$$ then the 2-class field tower of $$k$$ terminates at $$k^2$$. The authors are interested in searching for fields with finite 2-class field tower of length at least 3. With this motivation they study the 2-rank of $$Cl_2(k^1)$$, and they determine the imaginary quadratic fields $$k$$ such that $$Cl_2(k^1)$$ are cyclic.
Since one of the authors [F. Lemmermeyer, J. Théor. Nombres Bordx. 6, 261-272 (1994; Zbl 0826.11052)] determined the cases where $$\text{Gal} (k^2/k)$$ are abelian or metacyclic, in this paper they treat the non-metacyclic case. For this the authors first prove that if $$Cl_2(k^1)$$ is cyclic, then $$Cl_2(k)\subseteq (2,2^m)$$ using Schur multipliers. Next the group structure of non-metacyclic 2-group $$G$$ such that $$G^{ab}\simeq(2,2^m)$$ with $$m>1$$ is studied in detail. Using this result they finally characterize the imaginary quadratic fields $$k$$ such that $$Cl_2(k^1)$$ are cyclic in terms of quadratic and quartic symbols about the ramified primes in $$k$$, and determine the group structure of $$\text{Gal} (k^2/k)$$.

### MSC:

 11R37 Class field theory 11R11 Quadratic extensions 11R29 Class numbers, class groups, discriminants

Zbl 0826.11052
Full Text:

### References:

 [1] Benjamin, E., Remarks concerning the 2-Hilbert class field of imaginary quadratic number fields, Bull. Austral. Math. Soc., 48, 379-383 (1993) · Zbl 0806.11047 [5] Blackburn, N., On prime-power groups in which the derived group has two generators, Proc. Cambridge Philos. Soc., 53, 19-27 (1957) · Zbl 0077.03202 [6] Blackburn, N., On prime power groups with two generators, Proc. Cambridge Philos. Soc., 54, 327-337 (1958) · Zbl 0083.01902 [7] Furtwängler, Ph., Über das Verhalten der Ideale des Grundkörpers im Klassenkörper, Monatsh. Math. Phys., 27, 1-15 (1916) · JFM 46.0246.01 [8] Hall, M.; Senior, J. K., The Groups of Order $$2^n(n (1964)$$, Macmillan: Macmillan New York [9] Hall, P., A contribution to the theory of Groups of prime power order, Proc. London Math. Soc., 36, 29-95 (1933) · Zbl 0007.29102 [10] Huppert, B., Endliche Gruppen I (1967), Springer-Verlag: Springer-Verlag Heidelberg · Zbl 0217.07201 [11] Jehne, W., On knots in algebraic number theory, J. Reine Angew. Math., 311/312, 215-254 (1979) · Zbl 0432.12006 [12] Kaplan, P., Sur le 2-groupe des classes d’idéaux des corps quadratiques, J. Reine Angew. Math., 283/284, 313-363 (1974) · Zbl 0337.12003 [13] Karpilovsky, G., Schur Multipliers. Schur Multipliers, London Math. Soc. Monographs (1987), Oxford Univ. Press: Oxford Univ. Press Oxford [14] Kisilevsky, H., Number fields with class number congruent to 4 mod 8 and Hilbert’s Theorem 94, J. Number Theory, 8, 271-279 (1976) · Zbl 0334.12019 [15] Kubota, T., Über den bizyklischen biquadratischen Zahlkörper, Nagoya Math. J., 10, 65-85 (1956) · Zbl 0074.03001 [17] Lemmermeyer, F., Unramified quaternion extensions of quadratic number fields, J. Théor. Nombres Bordeaux, 9, 51-68 (1997) · Zbl 0890.11031 [18] Lemmermeyer, F., Ideal class groups of cyclotomic number fields, Acta Arith., 72.4, 347-359 (1995) · Zbl 0837.11059 [19] Lemmermeyer, F., On 2-class field towers of imaginary quadratic number fields, J. Théor. Nombres Bordeaux, 6, 261-272 (1994) · Zbl 0826.11052 [20] Lorenz, F., Algebraische Zahlentheorie (1993), Bibl. Inst. Mannheim: Bibl. Inst. Mannheim Mannheim [21] Miyake, K., Algebraic investigations of Hilbert’s Theorem 94, the principal ideal theorem, and the capitulation problem, Expo. Math., 7, 289-346 (1989) · Zbl 0704.11048 [22] Rédei, L.; Reichardt, H., Die Anzahl der durch 4 teilbaren Invarianten der Klassengruppe eines beliebigen quadratischen Zahlkörpers, J. Reine Angew. Math., 170, 69-74 (1933) · Zbl 0007.39602 [23] Sag, T. W.; Wamsley, J., Minimal presentations for groups of order $$2^n,n$$, J. Austral. Math., 15, 461-469 (1973) · Zbl 0267.20028 [24] Scholz, A., Totale Normenreste, die keine Normen sind, als Erzeuger nichtabelscher Körpererweiterungen, I, J. Reine Angew. Math., 175, 100-107 (1936) · JFM 62.0169.01 [25] Scholz, A., Totale Normenreste, die keine Normen sind, als Erzeuger nichtabelscher Körpererweiterungen, II, J. Reine Angew. Math., 182, 217-234 (1940) · JFM 66.0124.01 [26] Schur, I., Untersuchungen über die Darstellungen der endlichen Gruppen durch gebrochene lineare Substitutionen, J. Reine Angew. Math., 132, 85-137 (1907) · JFM 38.0174.02 [27] Serre, J.-P., Sur une question d’Olga Taussky, J. Number Theory, 2, 235-236 (1970) · Zbl 0209.05203 [28] Tate, J., Global class field theory, (Cassels; Fröhlich, Algebraic Number Theory (1967), Academic Press: Academic Press London and New York), 162-203 · Zbl 1179.11041 [29] Taussky, O., A remark on the class field tower, J. London Math. Soc., 12, 82-85 (1937) · JFM 63.0144.03 [30] Wiegold, J., The Schur multiplier: An elementary approach, (Campbell, C. M.; Robertson, E. F., Groups: St. Andrews, 1981. Groups: St. Andrews, 1981, London Math. Soc. Lecture Note Series, 71 (1982), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 137-154 · Zbl 0502.20003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.