On sums and products in a field

Zhou, Guang-Liang ; Sun, Zhi-Wei

Czechoslovak Mathematical Journal, Tome 72 (2022) no. 3, pp. 817-823

Résumé

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 $a_1+\dots +a_{2k}=x=a_1\dots a_{2k}$.

Détail

DOI : 10.21136/CMJ.2021.0184-21

Classification : 11D85, 11P99, 11T99
Keywords: field; rational function; restricted sum; restricted product