Genus theory, governing field, ramification and Frobenius
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910512972824576 |
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| author | Mfumu, Roslan Ibara Ngiza Maire, Christian |
| author_facet | Mfumu, Roslan Ibara Ngiza Maire, Christian |
| contents | In this work we develop, through a governing field, genus theory for a number field $\K$ with tame ramification in $T$ and splitting in $S$, where $T$ and $S$ are finite disjoint sets of primes of $\K$. This approach extends that initiated by the second author in the case of the class group. It allows expressing the $S$-$T$ genus number of a cyclic extension $Ł/\K$ of degree $p$ in terms of the rank of a matrix constructed from the Frobenius elements of the primes ramified in $Ł/\K$, in the Galois group of the underlying governing extension. For quadratic extensions $Ł/\Q$, the matrices in question are constructed from the Legendre symbols between the primes ramified in $Ł/\Q$ and the primes in $S$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_03754 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Genus theory, governing field, ramification and Frobenius Mfumu, Roslan Ibara Ngiza Maire, Christian Number Theory In this work we develop, through a governing field, genus theory for a number field $\K$ with tame ramification in $T$ and splitting in $S$, where $T$ and $S$ are finite disjoint sets of primes of $\K$. This approach extends that initiated by the second author in the case of the class group. It allows expressing the $S$-$T$ genus number of a cyclic extension $Ł/\K$ of degree $p$ in terms of the rank of a matrix constructed from the Frobenius elements of the primes ramified in $Ł/\K$, in the Galois group of the underlying governing extension. For quadratic extensions $Ł/\Q$, the matrices in question are constructed from the Legendre symbols between the primes ramified in $Ł/\Q$ and the primes in $S$. |
| title | Genus theory, governing field, ramification and Frobenius |
| topic | Number Theory |
| url | https://arxiv.org/abs/2407.03754 |