In this paper, we investigate an inverse problem on the Gauss sums of characters of finite fields. Namely, given a nontrivial additive character ψ, for two multiplicative characters ${\chi }_1$ and ${\chi }_2$ of ${\mathbb{F}}_{q^n}^{\times }$, if the Gauss sums $G({\chi }_1\cdot \eta ,\psi )=G({\chi }_2\cdot \eta ,\psi )$ for all characters η of ${\mathbb{F}}_q^{\times }$, when is ${\chi }_1$ equal to ${\chi }_2$ up to a Frobenius twist? This paper proves that the answer is positive for regular characters when $n\le 5$, or for $n<\frac{q-1}{2\sqrt{q}}+1$ in the appendix by Zhiwei Yun. In addition, we conjectured that the answer is positive when n is prime.

中文翻译：

---

高斯和的逆定理

在本文中，我们研究关于有限域字符的高斯和的反问题。即，给定一个非平凡的加法字符ψ，对于两个乘法字符${\chi }_{\mathrm{1个}}$ 和 ${\chi }_2$ 的 ${\mathbb{F}}_{q^{ñ}}^{\times }$，如果高斯求和 $G（{\chi }_{\mathrm{1个}}\cdot \eta ，\psi ）=G（{\chi }_2\cdot \eta ，\psi ）$对所有字符η的${\mathbb{F}}_q^{\times }$， 什么时候 ${\chi }_{\mathrm{1个}}$ 等于 ${\chi }_2$达到Frobenius的转折？本文证明对于正则字符，答案是肯定的。$ñ\le 5$，或 $ñ<\frac{q-\mathrm{1个}}{2\sqrt{q}}+\mathrm{1个}$在附录中由志伟云。另外，我们推测当n为质数时答案为肯定。