Let $\mathbb F_q$ be the finite field of order $q$, where $q\equiv 1\pmod 4$ is some prime power. Let $\chi_4\colon\mathbb F_q^\times\to\mathbb C^\times$ be a multiplicative character of exact order 4 (so $\chi_4(\theta)=i$ for some generator $\theta$ of the cyclic multiplicative group $\mathbb F_q^\times$). In Neal Koblitz's "Introduction to Elliptic Curves and Modular Forms," Exercise 3 in Section II.2 says to show that $\chi_4(-4)=1$. To do this, it suggests showing that $\chi_4(4)=\chi_4(-1)=1$ if $q\equiv 1\pmod 8$ and $\chi_4(4)=\chi_4(-1)=-1$ if $q\equiv 5\pmod 8$.
I first tried proving that $\chi_4(4)=1$ if $q\equiv 1\pmod 8$. It would be nice if I could prove that $2=u^2$ for some $u\in\mathbb F_q^\times$ so that I could say $\chi_4(4)=\chi_4(u)^4=1$, but I got stuck with this. It dawned on me that I really don't know how to do computations in finite fields. How can I prove the given problem. More generally, is there any nice technique for handling computations like this?