Geodesic
On sums and products in a field
Czechoslovak Mathematical Journal, Tome 72 (2022) no. 3, pp. 817-823
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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 $a_1+\dots +a_{2k}=x=a_1\dots a_{2k}$.
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 $a_1+\dots +a_{2k}=x=a_1\dots a_{2k}$.
DOI : 10.21136/CMJ.2021.0184-21
Classification : 11D85, 11P99, 11T99
Keywords: field; rational function; restricted sum; restricted product 
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Zhou, Guang-Liang; Sun, Zhi-Wei. On sums and products in a field. Czechoslovak Mathematical Journal, Tome 72 (2022) no. 3, pp. 817-823. doi: 10.21136/CMJ.2021.0184-21

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