# Journal of the Ramanujan Mathematical Society

Volume 30, Issue 4, December 2015 pp. 413–454.

l-Class groups of cyclic extensions of prime degree l

**Authors**:
Manisha Kulkarni, Dipramit Majumdar and Balasubramanian Sury

**Author institution:**Department of Mathematics, International Institute of Information Technology, 26/C, Electronics City, Hosur Road, Bangalore 560 100, India

**Summary: **
Let K/F be a cyclic extension of odd prime degree l over a
number field F. If F has class number coprime to l, we study
the structure of the l-Sylow subgroup of the class group of K.
In particular, when F contains the l-th roots of unity, we
obtain bounds for the Fl-rank of the l-Sylow
subgroup of K using genus theory. We obtain some results valid
for general l. Following that, we obtain more complete, explicit
results for l = 5 and F = Q(e(2iπ/5)). The
rank of the 5-class group of K is expressed in terms of
power residue symbols. We compare our results with tables obtained
using SAGE (the latter is under GRH). We obtain explicit results
in several cases. These results have a number of potential
applications. For instance, some of them like Theorem 5.16 could
be useful in the arithmetic of elliptic curves over towers of the
form Q(e(2iπ/5n, x(1/5)). Using the
results on the class groups of the fields of the form
Q(e(2iπ/5), x(1/5)), and using Kummer
duality theory, we deduce results on the 5-class numbers of
fields of the form Q(x(1/5)).

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