| 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 |
| . |
| Category:
|
math |
| . |
| 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 |
| . |
| Date available:
|
2022-08-22T08:24:17Z |
| Last updated:
|
2024-10-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/150619 |
| . |
| 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. |
| . |
