Solving equations over finite fields is an important problem from both theoretical and practice points of view. The problem of solving explicitly the equation $P_a(X)=0$ over the finite field ${\mathbb{F}}_Q$, where $P_a(X):=X^{q+1}+X+a$, $Q=p^n$, $q=p^k$, $a\in {\mathbb{F}}_Q^{⁎}$ and p is a prime, arises in many different contexts including finite geometry, the inverse Galois problem [1], the construction of difference sets with Singer parameters [9], determining cross-correlation between m-sequences [10] and to construct error correcting codes [5], cryptographic APN functions [6], [7], designs [21], as well as to speed up the index calculus method for computing discrete logarithms on finite fields [11], [12] and on algebraic curves [18].

In fact, the research on this specific problem has a long history of more than a half-century from the year 1967 when Berlekamp, Rumsey and Solomon [2] firstly considered a very particular case with $k=1$ and $p=2$.

In this article, we discuss the equation $P_a(X)=0$ without any restriction on p and $\gcd (n,k)$. In a very recent paper [15], the authors have left open a problem that could definitely solve this equation. More specifically, for the cases of one or two ${\mathbb{F}}_Q$-zeros, explicit expressions for these rational zeros in terms of a were provided, but for the case of $p^{\gcd (n,k)}+1$ ${\mathbb{F}}_Q-$ zeros it was remained open to compute explicitly the zeros. This paper solves the remained problem, thus now the equation $X^{p^k+1}+X+a=0$ over ${\mathbb{F}}_{p^n}$ is completely solved for any prime p, any integers n and k.

从理论和实践的角度来看，求解有限域上的方程都是一个重要的问题。显式求解方程的问题${磷}_{\mathrm{一种}}(X)=0$ 在有限域上 ${\mathbb{F}}_{问}$， 在哪里 ${磷}_{\mathrm{一种}}(X)：=X^{q+1}+X+\mathrm{一种}$, $问={磷}^n$, $q={磷}^{克}$, $\mathrm{一种}\in {\mathbb{F}}_{问}^{⁎}$和p是素数，产生于许多不同的环境，包括有限的几何形状，逆伽罗瓦问题[1]，的差集的结构与歌手参数[9]，确定之间的互相关米-sequences [10]，并构建错误校正代码 [5]、加密 APN 函数 [6]、[7]、设计 [21]，以及加快指数演算方法，用于计算有限域 [11]、[12] 和代数曲线上的离散对数[18]。

事实上，从 1967 年 Berlekamp、Rumsey 和 Solomon [2] 首次考虑一个非常特殊的情况开始，对这个特定问题的研究已有半个多世纪的悠久历史。 $克=1$ 和 $磷=2$.

在本文中，我们讨论方程 ${磷}_{\mathrm{一种}}(X)=0$对p和$\mathrm{GCD}(n,克)$. 在最近的一篇论文 [15] 中，作者留下了一个绝对可以解决这个方程的问题。更具体地说，对于一两个的情况${\mathbb{F}}_{问}$-zeros，对于在这些方面理性零明确表达一个被提供，但是的情况下${磷}^{\mathrm{GCD}(n,克)}+1$ ${\mathbb{F}}_{问}-$zeros 它仍然开放以明确计算零点。本文解决了剩下的问题，因此现在方程$X^{{磷}^{克}+1}+X+\mathrm{一种}=0$ 超过 ${\mathbb{F}}_{{磷}^n}$对任何素数p、任何整数n和k完全求解。