Exercise 8.7

Proof. As $\chi$ is a character of order 4, ${\chi }^2$ is a character of order 2, and $\rho$ (Legendre’s character) is the unique character of order 2, so ${\chi }^2=\rho$.

From Prop. 8.3.3 we have

Squaring the result of Ex. 8.5, we obtain

Moreover $\chi (2^4)={\chi }^4(2)=𝜀(2)=1$, and $g({\chi }^2)=g(\rho )=g$, so ${\left[g({\chi }^2)\right]}^2=g^2={(-1)}^{(p-1)∕2}p=p$ (From Prop. 6.3.2 and $p\equiv 1(\mathit{mod}4))$.

Equating these two results, we obtain

As $g{(\chi )}^4\ne 0$ since $|g(\chi ){\text{|}}^2=p$, we have $J(\chi ,\rho )\ne 0$, so

${[\chi (-1)]}^2=\chi ({(-1)}^2)=\chi (1)=1$, so $\chi (-1)=\pm 1$, and $\chi {(-1)}^{-1}=\chi (-1)$, thus

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