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On l-class groups of certain number fields

Published online by Cambridge University Press:  26 February 2010

Frank Gerth III
Frank Gerth III
Affiliation:
Department of Mathematics, University of Texas, Austin, Texas 78712, U. S. A.
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Let M be a finite extension field of the rational numbers Q, and let CM denote the ideal class group of M. Let l be a rational prime, and let AM denote the l-class group of M (i.e., the Sylow l-subgroup of CM). If G is any finite abelian l-group, we define

where ℤl is the ring of l-adic integers and Fl is the finite field of l elements. One of the classical results of algebraic number theory is the specification of rank AM when M is a quadratic extension of Q and l = 2. This result was obtained by means of Gauss's theory of genera. A generalization of this result can be found in [1], where A. Fröhlich has obtained upper and lower bounds for rank AM when M is a cyclic extension of Q of degree l. His methods also show how to compute rank AM exactly when l = 3. In [4], G. Gras has described a procedure for analyzing the l-class groups of relatively cyclic extensions of degree l. However when l > 3, the computations can be very difficult.

Type
Research Article
Information
Mathematika , Volume 23 , Issue 1 , June 1976 , pp. 116 - 123
Copyright
Copyright © University College London 1976

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References

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