Class field towers of imaginary quadratic fields. (English) Zbl 0611.12009
Let \(F\) be an imaginary quadratic field whose ideal class group has an elementary abelian Sylow 3-subgroup of order \(3^ 2\). A. Scholz and O. Taussky [J. Reine Angew. Math. 171, 19–41 (1934; Zbl 0009.10202)] proved that if \(F\) has a certain capitulation type then the 3-class field tower of \(F\) is of length 2. The present authors prove this in a simpler way. For another particular capitulation type they show, by constructing a group theoretic example, that it is possible for the 3-class field tower to be of length 3. This corrects an assertion made by Scholz and Taussky [op. cit.].
Reviewer: Tauno Metsänkylä (Turku)
MSC:
| 11R37 | Class field theory |
| 11R11 | Quadratic extensions |
| 11R29 | Class numbers, class groups, discriminants |
Citations:
Zbl 0009.10202Online Encyclopedia of Integer Sequences:
Absolute discriminants of complex quadratic fields with 3-class group of elementary abelian type (3,3) of rank 2.Absolute discriminants of complex quadratic fields with 3-class group of type (3,3) and Hilbert 3-class field tower of exact length 3, except for the cases mentioned in the COMMENTS.
References:
| [1] | Blackburn, N.: On a Special Class of p-Groups. Acta Math.100, 45-92 (1958). · Zbl 0083.24802 · doi:10.1007/BF02559602 |
| [2] | Brink, James R.: The Class Field Tower of Imaginary Quadratic Number Fields of Type (3,3). Ohio State University, Ph.D. Dissertation, 1984. |
| [3] | Hall, Marshall: The Theory of Groups. MacMillan, New York, 1959. · Zbl 0084.02202 |
| [4] | Scholz, A. and Taussky, O.: Die Hauptideale der kubischen Klassenkörper imaginär-quadratisher Zahlkörper. J. f. Math171, 19-41 (1934). · Zbl 0009.10202 |
| [5] | Serre, J. P.: Corps Locaux. Hermann, Paris, 1968. |
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