Let $\pi = a+bi \in \mathbb{Z}[i]$ and $q \equiv 3 \pmod{4}$ a rational prime. Show that $\pi^q \equiv \bar{\pi} \pmod{q}.$
It's a problem from chapter 9 "cubic and biquadratic reciprocity" of Rosen's classical introduction to modern number theory, and I have no idea how to solve, please helps.
Best Answer
If you raise $a+bi$ to the $q$-th power and use binomial theorem all the terms are divisible by $q$ except the first and the last so $$ (a+bi)^q \equiv a^q + b^q i^q \pmod{q}$$ Now you can use Fermat's little theorem and the periodicity of the power's of $i$ to find the result.