This paper deals with the following problem. Given a finite extension of fields $\mathbb{L}/\mathbb{K}$ and denoting the trace map from $\mathbb{L}$ to $\mathbb{K}$ by $\text{Tr}$ , for which elements $z$ in $\mathbb{L}$ , and $a$ , $b$ in $\mathbb{K}$ , is it possible to write $z$ as a product $xy$ , where $x,y\in \mathbb{L}$ with $\text{Tr}(x)=a,\text{Tr}(y)=b$ ? We solve most of these problems for finite fields, with a complete solution when the degree of the extension is at least 5. We also have results for arbitrary fields and extensions of degrees 2, 3 or 4. We then apply our results to the study of perfect nonlinear functions, semifields, irreducible polynomials with prescribed coefficients, and a problem from finite geometry concerning the existence of certain disjoint linear sets.

本文处理以下问题。给定域$\mathbb{L}/\mathbb{K}$的有限扩展，并用$\text{Tr}$表示从$\mathbb{L}$到$\mathbb{K}$的轨迹映射，对于$\mathbb{L}$中的$z$和$\mathbb{K}$中的 $a $和$b$中的哪些元素，是否可以将$z$写为乘积$xy$，其中$x， y\in \mathbb{L}$与$\text{Tr}(x)=a,\text{Tr}(y)=b$ ? 我们解决了有限域的大部分问题，当扩展度至少为 5 时，我们得到了完整的解决方案。我们还得到了任意域和 2、3 或 4 度扩展的结果。然后，我们将结果应用于研究完美非线性函数，半场，具有规定系数的不可约多项式，以及有限几何中有关某些不相交线性集存在的问题。