Exercise 1.16

Proof. Here $u,v\in \mathbb{N}$, where $\mathbb{N}=\{0,1,2,\dots \}$.

For all primes $p$ such that $p\mid u$, ${\mathrm{ord}}_p(u)+{\mathrm{ord}}_p(v)=2{\mathrm{ord}}_p(a)$. As $u\wedge v=1$ and $p\mid u$, then $p\nmid v$, thus ${\mathrm{ord}}_p(v)=0$. Therefore ${\mathrm{ord}}_p(u)$ is even for all prime $p$ such that $p\mid u$. From Exercise 1.15, we can conclude that $u$ is a square. Similarly, $v$ is a square. □