Czechoslovak Mathematical Journal, Vol. 72, No. 3, pp. 817-823, 2022
On sums and products in a field
Guang-Liang Zhou, Zhi-Wei Sun
Received May 14, 2021. Published online November 25, 2021.
Abstract: We study sums and products in a field. Let $F$ be a field with ${\rm ch}(F)\not=2$, where ${\rm\ch} (F)$ is the characteristic of $F$. For any integer $k\gs4$, 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 $a_1+\dots+a_{2k}=x=a_1\dots a_{2k}$.
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Affiliations: Guang-Liang Zhou (corresponding author), Zhi-Wei Sun, Department of Mathematics, Nanjing University, Gulou Campus, No. 22, Hankou Road, Gulou District, Nanjing 210093, P. R. China, e-mail: guangliangzhou@126.com, zwsun@nju.edu.cn
