Exercise 9.3

Let $\gamma$ a primary prime. To evaluate ${\chi }_{\gamma }(\mu )$ we see, by Exercise 2, that it is enough to evaluate ${\chi }_{\gamma }(-1),{\chi }_{\gamma }(\omega ),{\chi }_{\gamma }(\lambda )$, and ${\chi }_{\gamma }(\pi )$, where $\pi$ is a primary prime. Since $-1={(-1)}^3$ we have ${\chi }_{\gamma }(-1)=1$. We now consider ${\chi }_{\gamma }(\omega )$. Let $\gamma =a+\mathit{b\omega }$ and set $a=3m-1$ and $b=3n$. Show that ${\chi }_{\gamma }(\omega )={\omega }^{m+n}$.