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INFINITE HILBERT CLASS FIELD TOWERS OVER CYCLOTOMIC FIELDS

Published online by Cambridge University Press:  01 January 2008

IGOR E. SHPARLINSKI
IGOR E. SHPARLINSKI*
Affiliation:
Department of Computing, Macquarie University, Sydney, NSW 2109, Australia e-mail: igor@ics.mq.edu.au
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Abstract

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We use a result of Y. Furuta to show that for almost all positive integers m, the cyclotomic field has an infinite Hilbert p-class field tower with high rank Galois groups at each step, simultaneously for all primes p of size up to about (log logm)1 + o(1). We also use a recent result of B. Schmidt to show that for infinitely many m there is an infinite Hilbert p-class field tower over for some p≥m0.3385 + o(1). These results have immediate applications to the divisibility properties of the class number of .

Keywords

11N2511R1711R37
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 50 , Issue 1 , January 2008 , pp. 27 - 32
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

REFERENCES

1.Baker, R. C. and Harman, G., Shifted primes without large prime factors, Acta Arith. 83 (1998), 331–361.CrossRefGoogle Scholar
2.Banks, W. D. and Shparlinski, I. E., On values taken by the largest prime factor of shifted primes, J. Aust. Math. Soc. 82 (2007), 133–147.CrossRefGoogle Scholar
3.Brumer, A., Ramification and class towers of number fields, Michigan Math. J. 12 (1965), 129–131.CrossRefGoogle Scholar
4.Cassels, J. W. S. and Froehlich, A., Algebraic number theory (Academic Press, London 1967).Google Scholar
5.Furuta, Y., On class field towers and the rank of ideal class groups, Nagoya Math. J. 48 (1972), 147–157.CrossRefGoogle Scholar
6.Gerth, F. III, On cyclic fields of odd prime degree p with infinite Hilbert p-class field towers, Canad. Math. Bull. 45 (2002), 86–88.CrossRefGoogle Scholar
7.Gerth, F. III, A density result for some imaginary quadratic fields with infinite Hilbert 2-class field towers, Arch. Math. (Basel) 82 (2004), 23–27.CrossRefGoogle Scholar
8.Hajir, F., On the growth of p-class groups in p-class field towers, J. Algebra 188 (1997), 256–271.CrossRefGoogle Scholar
9.Halberstam, H. and Richert, H.-E., Sieve methods (Academic Press, London, 1974).Google Scholar
10.Lemmermeyer, F., Ideal class groups of cyclotomic number fields, II, Acta Arith. 84 (1998), 59–70.CrossRefGoogle Scholar
11.Norton, K. K., On the number of restricted prime factors of an integer, I, Illinois J. Math. 20 (1976), 681–705.CrossRefGoogle Scholar
12.Pomerance, C., ‘On the distribution of amicable numbers’, J. reine angew. Math., 293/294 (1977), 217–222.Google Scholar
13.Prachar, K., Primzahlverteilung (Springer-Verlag, 1957).Google Scholar
14.Schmidt, B., The field descent and class groups of CM-fields, Acta Arith. 119 (2005), 291–306.CrossRefGoogle Scholar
15.Schoof, R., Infinite class field towers of quadratic fields, J. Reine Angew. Math. 372 (1986), 209–220.Google Scholar
16.Takeuchi, T., Notes on the class field towers of cyclic fields of degree l, Tôhoku Math. J. 31 (1979), 301–307.CrossRefGoogle Scholar