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We study sums and products in a field. Let $F$ be a field with $\mathrm{ch}\left(F\right)\ne 2$, where $\mathrm{ch}\left(F\right)$ is the characteristic of $F$. For any integer $k\ge 4$, we show that any $x\in F$ can be written as $a_1+\cdots +a_k$ with $a_1,\cdots ,a_k\in F$ and $a_1\cdots a_k=1$, and that for any $\alpha \in F\setminus \left\{0\right\}$ we can write every $x\in F$ as $a_1\cdots a_k$ with $a_1,\cdots ,a_k\in F$ and $a_1+\cdots +a_k=\alpha$. We also prove that for any $x\in F$ and $k\in \{2,3,\cdots \}$ there are $a_1,\cdots ,a_{2k}\in F$ such that $a_1+\cdots +a_{2k}=x=a_1\cdots a_{2k}$.

Let $l⩾2$ be an integer. Recently, Hu and Lü offered the asymptotic formula for the sum of the higher divisor function $$\sum _{1⩽n_1,n_2,...,n_l⩽x^{1/2}}{\tau }_k(n_1^2+n_2^2+\cdots +n_l^2),$$ where ${\tau }_k\left(n\right)$ represents the $k$th divisor function. We give the Goldbach-type analogy of their result. That is to say, we investigate the asymptotic behavior of the sum $$\sum _{1⩽p_1,p_2,...,p_l⩽x}{\tau }_k(p_1+p_2+\cdots +p_l),$$ where $p_1,p_2,\cdots ,p_l$ are prime variables.