Prime Splitting and Common Index Divisors in Radical Extensions
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arXiv
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866908462981578752 |
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| author | Smith, Hanson |
| author_facet | Smith, Hanson |
| contents | We explicitly describe the splitting of odd integral primes in the radical extension $\mathbb{Q}(\sqrt[n]{a})$, where $x^n-a$ is an irreducible polynomial in $\mathbb{Z}[x]$. Our motivation is to classify common index divisors, the primes whose splitting provides a local obstruction to the existence of a power integral basis for the ring of integers of $\mathbb{Q}(\sqrt[n]{a})$. Among other results, we show that if $p$ is such a prime, even or otherwise, then $p$ divides $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_08911 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Prime Splitting and Common Index Divisors in Radical Extensions Smith, Hanson Number Theory 11R04, 11R21, 11R27 We explicitly describe the splitting of odd integral primes in the radical extension $\mathbb{Q}(\sqrt[n]{a})$, where $x^n-a$ is an irreducible polynomial in $\mathbb{Z}[x]$. Our motivation is to classify common index divisors, the primes whose splitting provides a local obstruction to the existence of a power integral basis for the ring of integers of $\mathbb{Q}(\sqrt[n]{a})$. Among other results, we show that if $p$ is such a prime, even or otherwise, then $p$ divides $n$. |
| title | Prime Splitting and Common Index Divisors in Radical Extensions |
| topic | Number Theory 11R04, 11R21, 11R27 |
| url | https://arxiv.org/abs/2409.08911 |