Article
| Title: | On sums and products in a field (English) |
| Author: | Zhou, Guang-Liang |
| Author: | Sun, Zhi-Wei |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 72 |
| Issue: | 3 |
| Year: | 2022 |
| Pages: | 817-823 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We study sums and products in a field. Let $F$ be a field with ${\rm ch}(F)\not =2$, where ${\rm {\rm ch} } (F)$ is the characteristic of $F$. For any integer $k\geq 4$, we show that any $x\in F$ can be written as $a_1+\dots +a_k$ with $a_1,\dots ,a_k\in F$ and $a_1\dots a_k=1$, and that for any $\alpha \in F \setminus \{0\}$ we can write every $x\in F$ as $a_1\dots a_k$ with $a_1,\dots ,a_k\in F$ and $a_1+\dots +a_k=\alpha $. We also prove that for any $x\in F$ and $k\in \{2,3,\dots \}$ there are $a_1,\dots ,a_{2k}\in F$ such that $a_1+\dots +a_{2k}=x=a_1\dots a_{2k}$. (English) |
| Keyword: | field |
| Keyword: | rational function |
| Keyword: | restricted sum |
| Keyword: | restricted product |
| MSC: | 11D85 |
| MSC: | 11P99 |
| MSC: | 11T99 |
| idZBL: | Zbl 07584104 |
| idMR: | MR4467944 |
| DOI: | 10.21136/CMJ.2021.0184-21 |
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| Date available: | 2022-08-22T08:24:17Z |
| Last updated: | 2022-12-27 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/150619 |
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| Reference: | [1] Elkies, N. D.: On the areas of rational triangles or how did Euler (and how can we) solve $xyz(x+y+z)=a$?.Available at \let \relax\brokenlink{http://www.math.harvard.edu/ elkies/{euler_14t.pdf}} (2014), 50 pages. |
| Reference: | [2] Klyachko, A. A., Mazhuga, A. M., Ponfilenko, A. N.: Balanced factorisations in some algebras.Available at https://arxiv.org/abs/1607.01957 (2016), 4 pages. |
| Reference: | [3] Klyachko, A. A., Vassilyev, A. N.: Balanced factorisations.Available at https://arxiv.org/abs/1506.01571 (2015), 8 pages. MR 3593641 |
| Reference: | [4] Zypen, D. van der: Question on a generalisation of a theorem by Euler.Question 302933 at MathOverflow, June 16, 2018. Available at http://mathoverflow.net/questions/302933. |
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