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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2301.05700 |
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| _version_ | 1866918402210136064 |
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| author | Ferri, Fabio Johnston, Henri |
| author_facet | Ferri, Fabio Johnston, Henri |
| contents | Let $L/K$ be a Galois extension of number fields and let $G=\mathrm{Gal}(L/K)$. We show that under certain hypotheses on $G$, for a fixed prime number $p$, Leopoldt's conjecture at $p$ for certain proper intermediate fields of $L/K$ implies Leopoldt's conjecture at $p$ for $L$. We also obtain relations between the Leopoldt defects of intermediate extensions of $L/K$. By applying a result of Buchmann and Sands together with an explicit description of units and a special case of the above results, we show that given any finite set of prime numbers $\mathcal{P}$, there exists an infinite family $\mathcal{F}$ of totally real $S_{3}$-extensions of $\mathbb{Q}$ such that Leopoldt's conjecture for $F$ at $p$ holds for every $F \in \mathcal{F}$ and $p \in \mathcal{P}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_05700 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Applications of representation theory and of explicit units to Leopoldt's conjecture Ferri, Fabio Johnston, Henri Number Theory 11R23 (Primary) 11R27, 19A22, 20C15 (Secondary) Let $L/K$ be a Galois extension of number fields and let $G=\mathrm{Gal}(L/K)$. We show that under certain hypotheses on $G$, for a fixed prime number $p$, Leopoldt's conjecture at $p$ for certain proper intermediate fields of $L/K$ implies Leopoldt's conjecture at $p$ for $L$. We also obtain relations between the Leopoldt defects of intermediate extensions of $L/K$. By applying a result of Buchmann and Sands together with an explicit description of units and a special case of the above results, we show that given any finite set of prime numbers $\mathcal{P}$, there exists an infinite family $\mathcal{F}$ of totally real $S_{3}$-extensions of $\mathbb{Q}$ such that Leopoldt's conjecture for $F$ at $p$ holds for every $F \in \mathcal{F}$ and $p \in \mathcal{P}$. |
| title | Applications of representation theory and of explicit units to Leopoldt's conjecture |
| topic | Number Theory 11R23 (Primary) 11R27, 19A22, 20C15 (Secondary) |
| url | https://arxiv.org/abs/2301.05700 |