Quasifinite fields of prescribed characteristic and Diophantine dimension
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| Format: | Preprint |
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2023
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| _version_ | 1866929249616658432 |
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| author | Chipchakov, Ivan D. Paunov, Boyan |
| author_facet | Chipchakov, Ivan D. Paunov, Boyan |
| contents | Let $\mathbb{P}$ be the set of prime numbers, $\overline {\mathbb{P}}$ the union $\mathbb{P} \cup \{0\}$, and for any field $E$, let char$(E)$ be its characteristic, ddim$(E)$ the Diophantine dimension of $E$, $\mathcal{G}_{E}$ the absolute Galois group of $E$, and cd$(\mathcal{G}_{E})$ the Galois cohomological dimension $\mathcal{G}_{E}$. The research presented in this paper is motivated by the open problem of whether cd$(\mathcal{G}_{E}) \le {\rm ddim}(E)$. It proves the existence of quasifinite fields $Φ_{q}\colon q \in \mathbb{P}$, with ddim$(Φ_{q})$ infinity and char$(Φ_{q}) = q$, for each $q$. It shows that for any integer $m > 0$ and $q \in \overline {\mathbb{P}}$, there is a quasifinite field $Φ_{m,q}$ such that char$(Φ_{m,q}) = q$ and ddim$(Φ_{m,q}) = m$. This is used for proving that for any $q \in \overline {\mathbb{P}}$ and each pair $k$, $\ell \in (\mathbb{N} \cup \{0, \infty \})$ satisfying $k \le \ell $, there exists a field $E _{k, \ell ; q}$ with char$(E _{k, \ell ; q}) = q$, ddim$(E _{k, \ell ; q}) = \ell $ and cd$(\mathcal{G}_{E_{k, \ell ; q}}) = k$. Finally, we show that the field $E _{k, \ell ; q}$ can be chosen to be perfect unless $k = 0 \neq \ell $. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2303_04112 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Quasifinite fields of prescribed characteristic and Diophantine dimension Chipchakov, Ivan D. Paunov, Boyan Number Theory Rings and Algebras (2020): 11E76, 12J10 (primary), 11S15, 12G10 (secondary) Let $\mathbb{P}$ be the set of prime numbers, $\overline {\mathbb{P}}$ the union $\mathbb{P} \cup \{0\}$, and for any field $E$, let char$(E)$ be its characteristic, ddim$(E)$ the Diophantine dimension of $E$, $\mathcal{G}_{E}$ the absolute Galois group of $E$, and cd$(\mathcal{G}_{E})$ the Galois cohomological dimension $\mathcal{G}_{E}$. The research presented in this paper is motivated by the open problem of whether cd$(\mathcal{G}_{E}) \le {\rm ddim}(E)$. It proves the existence of quasifinite fields $Φ_{q}\colon q \in \mathbb{P}$, with ddim$(Φ_{q})$ infinity and char$(Φ_{q}) = q$, for each $q$. It shows that for any integer $m > 0$ and $q \in \overline {\mathbb{P}}$, there is a quasifinite field $Φ_{m,q}$ such that char$(Φ_{m,q}) = q$ and ddim$(Φ_{m,q}) = m$. This is used for proving that for any $q \in \overline {\mathbb{P}}$ and each pair $k$, $\ell \in (\mathbb{N} \cup \{0, \infty \})$ satisfying $k \le \ell $, there exists a field $E _{k, \ell ; q}$ with char$(E _{k, \ell ; q}) = q$, ddim$(E _{k, \ell ; q}) = \ell $ and cd$(\mathcal{G}_{E_{k, \ell ; q}}) = k$. Finally, we show that the field $E _{k, \ell ; q}$ can be chosen to be perfect unless $k = 0 \neq \ell $. |
| title | Quasifinite fields of prescribed characteristic and Diophantine dimension |
| topic | Number Theory Rings and Algebras (2020): 11E76, 12J10 (primary), 11S15, 12G10 (secondary) |
| url | https://arxiv.org/abs/2303.04112 |