On Ozaki's theorem realizing prescribed $p$-groups as $p$-class tower groups
Fuente:
arXiv
Salvato in:
| Autori principali: | , , |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2022
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909120535199744 |
|---|---|
| author | Hajir, Farshid Maire, Christian Ramakrishna, Ravi |
| author_facet | Hajir, Farshid Maire, Christian Ramakrishna, Ravi |
| contents | We give a streamlined and effective proof of Ozaki's theorem that any finite $p$-group $Γ$ is the Galois group of the $p$-Hilbert class field tower of some number field $\rm F$. Our work is inspired by Ozaki's and applies in broader circumstances. While his theorem is in the totally complex setting, we obtain the result in any mixed signature setting for which there exists a number field ${\rm k}_0$ with class number prime to $p$. We construct ${\rm F}/{\rm k}_0$ by a sequence of ${\mathbb Z}/p$-extensions ramified only at finite tame primes and also give explicit bounds on $[{\rm F}:{\rm k}_0]$ and the number of ramified primes of ${\rm F}/{\rm k}_0$ in terms of $\# Γ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2204_08408 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | On Ozaki's theorem realizing prescribed $p$-groups as $p$-class tower groups Hajir, Farshid Maire, Christian Ramakrishna, Ravi Number Theory 11R29 We give a streamlined and effective proof of Ozaki's theorem that any finite $p$-group $Γ$ is the Galois group of the $p$-Hilbert class field tower of some number field $\rm F$. Our work is inspired by Ozaki's and applies in broader circumstances. While his theorem is in the totally complex setting, we obtain the result in any mixed signature setting for which there exists a number field ${\rm k}_0$ with class number prime to $p$. We construct ${\rm F}/{\rm k}_0$ by a sequence of ${\mathbb Z}/p$-extensions ramified only at finite tame primes and also give explicit bounds on $[{\rm F}:{\rm k}_0]$ and the number of ramified primes of ${\rm F}/{\rm k}_0$ in terms of $\# Γ$. |
| title | On Ozaki's theorem realizing prescribed $p$-groups as $p$-class tower groups |
| topic | Number Theory 11R29 |
| url | https://arxiv.org/abs/2204.08408 |