On sums and products in a field

In this paper we study sums and products in a field. Let $F$ be a field with $\mathrm{c}\mathrm{h}(F)≠2$, where $\mathrm{c}\mathrm{h}(F)$ is the characteristic of $F$. For any integer $k\ge 4$, we show that each $x\in F$ can be written as $a_1+\dots +a_k$ with $a_1,\dots ,a_k\in F$ and $a_1\dots a_k=1$ if $\mathrm{c}\mathrm{h}(F)≠3$, and that for any $\alpha \in F\setminus \{0\}$ we can write each $x\in F$ 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 $a_1+\dots +a_{2k}=x=a_1\dots a_{2k}$.