Question 137
Consider a field $\mathbb{L}$ with $q$ elements. Prove that:
a) if $q \equiv 3 \pmod 4$ and $n \in \mathbb{N}$, $n \ge 2$, is a positive integer divisible by $q-1$, then $x^n = (x^2+1)^n$ for any $x \in \mathbb{L}^*$;
b) if there is an integer $n \in \mathbb{N}^*, n \ge 2$, such that $x^n = (x^2+1)^n$ for any $x \in \mathbb{L}^*$, then $q \equiv 3 \pmod 4$ and $q-1$ is a divisor of $n$.
a) if $q \equiv 3 \pmod 4$ and $n \in \mathbb{N}$, $n \ge 2$, is a positive integer divisible by $q-1$, then $x^n = (x^2+1)^n$ for any $x \in \mathbb{L}^*$;
b) if there is an integer $n \in \mathbb{N}^*, n \ge 2$, such that $x^n = (x^2+1)^n$ for any $x \in \mathbb{L}^*$, then $q \equiv 3 \pmod 4$ and $q-1$ is a divisor of $n$.