## Class groups of Kummer extensions via cup products in Galois cohomology

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- by Karl Schaefer and Eric Stubley PDF
- Trans. Amer. Math. Soc.
**372**(2019), 6927-6980 Request permission

## 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 $N$.## References

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## Additional Information

**Karl Schaefer**- Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois
- MR Author ID: 1131390
- Email: karl@math.uchicago.edu
**Eric Stubley**- Affiliation: Department of Mathematics, University of Chicago, Chicago, Illinois
- Email: stubley@uchicago.edu
- 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)
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**372**(2019), 6927-6980 - MSC (2010): Primary 11R29; Secondary 11R34
- DOI: https://doi.org/10.1090/tran/7746
- MathSciNet review: 4024543