Prime Splitting and Common $N$-Index Divisors in Radical Extensions: Part $p=2$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915698723258368 |
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| author | Scofield, Dylan Smith, Hanson |
| author_facet | Scofield, Dylan Smith, Hanson |
| contents | Following work of Vélez, we explicitly describe the splitting of the integral prime 2 in the radical extension $\mathbb{Q}(\sqrt[n]{a})$, where $x^n-a$ is an irreducible polynomial in $\mathbb{Z}[x]$. With previous work of the second author, this fully describes the splitting of any prime in $\mathbb{Q}(\sqrt[n]{a})$. Using this description, we classify common index divisors (the primes whose splitting prevents the existence of a power integral basis for the ring of integers). Using work of Pleasants, we extend this to describe common $N$-index divisors (primes that divide the index of any order generated over $\mathbb{Z}$ by $N$ elements). We also present two novel constructions of non-monogenic fields with no common index divisors as well as constructions of number rings requiring $N$ ring generators for any $N>1$. Examples are provided throughout. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_23677 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Prime Splitting and Common $N$-Index Divisors in Radical Extensions: Part $p=2$ Scofield, Dylan Smith, Hanson Number Theory 11R04, 11R21, 11R27 Following work of Vélez, we explicitly describe the splitting of the integral prime 2 in the radical extension $\mathbb{Q}(\sqrt[n]{a})$, where $x^n-a$ is an irreducible polynomial in $\mathbb{Z}[x]$. With previous work of the second author, this fully describes the splitting of any prime in $\mathbb{Q}(\sqrt[n]{a})$. Using this description, we classify common index divisors (the primes whose splitting prevents the existence of a power integral basis for the ring of integers). Using work of Pleasants, we extend this to describe common $N$-index divisors (primes that divide the index of any order generated over $\mathbb{Z}$ by $N$ elements). We also present two novel constructions of non-monogenic fields with no common index divisors as well as constructions of number rings requiring $N$ ring generators for any $N>1$. Examples are provided throughout. |
| title | Prime Splitting and Common $N$-Index Divisors in Radical Extensions: Part $p=2$ |
| topic | Number Theory 11R04, 11R21, 11R27 |
| url | https://arxiv.org/abs/2512.23677 |