Class groups of Kummer extensions via cup products in Galois cohomology
| Autor: | Eric Stubley, Karl Schaefer |
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| Rok vydání: | 2018 |
| Předmět: |
Class (set theory)
Mathematics - Number Theory Galois cohomology Group (mathematics) Applied Mathematics General Mathematics 010102 general mathematics Characterization (mathematics) 01 natural sciences Prime (order theory) Combinatorics Converse FOS: Mathematics Number Theory (math.NT) 0101 mathematics Mathematics Counterexample |
| DOI: | 10.48550/arxiv.1806.00517 |
| Popis: | 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 $N$. Comment: 54 pages. This version has been accepted for publication in Transaction of the AMS |
| Databáze: | OpenAIRE |
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