The Proportion of Irreducible p-adic Polynomials
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866916655721873408 |
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| author | Rajagopal, Isaac |
| author_facet | Rajagopal, Isaac |
| contents | We attempt to quantify the exact proportion of monic $p$-adic polynomials of degree $n$ which are irreducible. We find an exact answer to this when $n$ is prime and $p \neq n$, and also when $n = 4$ and $p \neq 2$. Our answers are rational functions in $p$. This relates to previous work done to find exact proportions of $p$-adic polynomials of degree $n$ which have $k$ roots. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_10534 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Proportion of Irreducible p-adic Polynomials Rajagopal, Isaac Number Theory We attempt to quantify the exact proportion of monic $p$-adic polynomials of degree $n$ which are irreducible. We find an exact answer to this when $n$ is prime and $p \neq n$, and also when $n = 4$ and $p \neq 2$. Our answers are rational functions in $p$. This relates to previous work done to find exact proportions of $p$-adic polynomials of degree $n$ which have $k$ roots. |
| title | The Proportion of Irreducible p-adic Polynomials |
| topic | Number Theory |
| url | https://arxiv.org/abs/2408.10534 |