MathNet 2011 Question 49

Consider $n \in \mathbb{N}$, $n \ge 2$, and $A$ a unitary ring with $n$ elements, such that the equation $x^{n+1} + x = 0$ has in $A \setminus \{0\}$ the unique solution $x = 1$. Prove that $A$ is a field.

Ioan Băetu