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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Class groups of Kummer extensions via cup products in Galois cohomology


Authors: Karl Schaefer and Eric Stubley
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 11R29; Secondary 11R34
DOI: https://doi.org/10.1090/tran/7746
Published electronically: May 30, 2019
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Abstract: We use Galois cohomology to study the $ p$-rank of the class group of $ \mathbf {Q}(N^{1/p})$, where $ N \equiv 1 \bmod {p}$ is prime. We prove a partial converse to a theorem of Calegari-Emerton, and provide a new explanation of the known counterexamples to the full converse of their result. In the case $ p = 5$, we prove a complete characterization of the $ 5$-rank of the class group of $ \mathbf {Q}(N^{1/5})$ in terms of whether or not $ \prod _{k=1}^{(N-1)/2} k^{k}$ and $ \frac {\sqrt {5} - 1}{2}$ are $ 5$th powers mod $ N$.


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Additional Information

Karl Schaefer
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois
Email: karl@math.uchicago.edu

Eric Stubley
Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois
Email: stubley@uchicago.edu

DOI: https://doi.org/10.1090/tran/7746
Received by editor(s): July 23, 2018
Received by editor(s) in revised form: October 2, 2018, and October 29, 2018
Published electronically: May 30, 2019
Additional Notes: The second author wishes to acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC)
Article copyright: © Copyright 2019 American Mathematical Society