1. CJM 2006 (vol 58 pp. 580)
|Annihilators for the Class Group of a Cyclic Field of Prime Power Degree, II |
We prove, for a field $K$ which is cyclic of odd prime power degree over the rationals, that the annihilator of the quotient of the units of $K$ by a suitable large subgroup (constructed from circular units) annihilates what we call the non-genus part of the class group. This leads to stronger annihilation results for the whole class group than a routine application of the Rubin--Thaine method would produce, since the part of the class group determined by genus theory has an obvious large annihilator which is not detected by that method; this is our reason for concentrating on the non-genus part. The present work builds on and strengthens previous work of the authors; the proofs are more conceptual now, and we are also able to construct an example which demonstrates that our results cannot be easily sharpened further.
Categories:11R33, 11R20, 11Y40
2. CJM 1997 (vol 49 pp. 283)
|The $2$-rank of the class group of imaginary bicyclic biquadratic fields |
A formula is obtained for the rank of the $2$-Sylow subgroup of the ideal class group of imaginary bicyclic biquadratic fields. This formula involves the number of primes that ramify in the field, the ranks of the $2$-Sylow subgroups of the ideal class groups of the quadratic subfields and the rank of a $Z_2$-matrix determined by Legendre symbols involving pairs of ramified primes. As applications, all subfields with both $2$-class and class group $Z_2 \times Z_2$ are determined. The final results assume the completeness of D.~A.~Buell's list of imaginary fields with small class numbers.
Categories:11R16, 11R29, 11R20