Past exam of the number theory course of the University of Oxford / 2011 / A / 1 / b / iii / Solution

Over the finite field ${\mathbb{F}}_7$,The nonzero squares modulo $7$ are $1,2,4$, so $-1$ is not a square. Adjoining a root of $x^2+1$ produces ${\mathbb{F}}_{49}$ and already gives both roots. Its Frobenius automorphism has order $2$ and interchanges them, whence