Exercise 8.19

Proof. Let $\chi$ be the Legendre character. Then

(We used Proposition 8.5.1). For all $k$, ${\chi }^{2k}=𝜀,{\chi }^{2k+1}=\chi$.

$\text{∙}$ If $r$ is odd, ${\chi }^r\ne 𝜀$, so $J_0(\chi ,\chi ,\cdots ,\chi )=0$ (Proposition 8.5.1).

$\text{∙}$ If $r$ is even, ${\chi }^r=𝜀$, so $J_0(\chi ,\chi ,\cdots ,\chi )=\chi (-1)(p-1)J(\chi ,\chi ,\cdots ,\chi )$, where there are $r-1$ components in the Jacobi sum ( Proposition 8.5.1).

By Theorem 3, $J(\chi ,\chi ,\cdots ,\chi )g({\chi }^{r-1})=g{(\chi )}^{r-1}$, and $g({\chi }^{r-1})=g(\chi )$, so

$g{(\chi )}^2=\chi (-1)p$, therefore $g{(\chi )}^{r-2}=\chi {(-1)}^{r∕2-1}{p}^{r∕2-1}={(-1)}^{((p-1)∕2)(r∕2-1)}{p}^{(r∕2-1)}$. So

(Verified in C++ with small values of $p$ and $r$.)

Conclusion :

□