Exercise 8.17

Proof. Let $g$ a generator in ${𝔽}_p^{\text{∗}}$. As ${(g^{(p-1)∕2})}^2=1$ and $g^{(p-1)∕2}\ne 1$, then $g^{(p-1)∕2}=-1$. As in Exercise 8.16, write $\chi (g)=e^{\mathit{qi\pi }∕2}$, with $q$ odd.

Then $-1$ is a fourth power in ${𝔽}_p^{\text{∗}}$ iff (see Exercice 8.16)

By Exercise 8.7, as $\chi$ is a character of order 4,

$\text{∙}$ If $p\equiv 1[8],$

$\chi (-1)=1$, so $J(\chi ,\chi )=J(\chi ,\rho )$, and ${\delta }_4(-1)=1$.

$\text{∙}$ If $p\equiv 5[8],$

$\chi (-1)=-1$, so $J(\chi ,\chi )=-J(\chi ,\rho )$, and ${\delta }_4(-1)=0$.