On sums and products in a field

Guang-Liang Zhou; Zhi-Wei Sun

Czechoslovak Mathematical Journal (2022)

  • Volume: 72, Issue: 3, page 817-823
  • ISSN: 0011-4642

Abstract

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We study sums and products in a field. Let F be a field with ch ( F ) ≠ 2 , where ch ( F ) is the characteristic of F . For any integer k ≥ 4 , we show that any x ∈ F can be written as a 1 + ⋯ + a k with a 1 , ⋯ , a k ∈ F and a 1 ⋯ a k = 1 , and that for any α ∈ F ∖ { 0 } we can write every x ∈ F as a 1 ⋯ a k with a 1 , ⋯ , a k ∈ F and a 1 + ⋯ + a k = α . We also prove that for any x ∈ F and k ∈ { 2 , 3 , ⋯ } there are a 1 , ⋯ , a 2 k ∈ F such that a 1 + ⋯ + a 2 k = x = a 1 ⋯ a 2 k .

How to cite

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Zhou, Guang-Liang, and Sun, Zhi-Wei. "On sums and products in a field." Czechoslovak Mathematical Journal 72.3 (2022): 817-823. <http://eudml.org/doc/298380>.

@article{Zhou2022,
abstract = {We study sums and products in a field. Let $F$ be a field with $\{\rm ch\}(F)\ne 2$, where $\{\rm \{\rm ch\} \} (F)$ is the characteristic of $F$. For any integer $k\ge 4$, 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 \lbrace 0\rbrace $ 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 \lbrace 2,3,\dots \rbrace $ there are $a_1,\dots ,a_\{2k\}\in F$ such that $a_1+\dots +a_\{2k\}=x=a_1\dots a_\{2k\}$.},
author = {Zhou, Guang-Liang, Sun, Zhi-Wei},
journal = {Czechoslovak Mathematical Journal},
keywords = {field; rational function; restricted sum; restricted product},
language = {eng},
number = {3},
pages = {817-823},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On sums and products in a field},
url = {http://eudml.org/doc/298380},
volume = {72},
year = {2022},
}

TY - JOUR
AU - Zhou, Guang-Liang
AU - Sun, Zhi-Wei
TI - On sums and products in a field
JO - Czechoslovak Mathematical Journal
PY - 2022
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 72
IS - 3
SP - 817
EP - 823
AB - We study sums and products in a field. Let $F$ be a field with ${\rm ch}(F)\ne 2$, where ${\rm {\rm ch} } (F)$ is the characteristic of $F$. For any integer $k\ge 4$, 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 \lbrace 0\rbrace $ 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 \lbrace 2,3,\dots \rbrace $ there are $a_1,\dots ,a_{2k}\in F$ such that $a_1+\dots +a_{2k}=x=a_1\dots a_{2k}$.
LA - eng
KW - field; rational function; restricted sum; restricted product
UR - http://eudml.org/doc/298380
ER -

References

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  1. Elkies, N. D., On the areas of rational triangles or how did Euler (and how can we) solve x y z ( x + y + z ) = a ?, Available at http://www.math.harvard.edu/ elkies/euler14t.pdf (2014), 50 pages. (2014) 
  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. (2016) 
  3. Klyachko, A. A., Vassilyev, A. N., Balanced factorisations, Available at https://arxiv.org/abs/1506.01571 (2015), 8 pages. (2015) MR3593641
  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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