Lecture 33.8 $\eta(x)= \begin{cases}1 & \text { if } x \text

3.8 $\eta(x)= \begin{cases}1 & \text { if } x \text { is a square in } \mathbb{F}_q^* \\ -1 & \text { if } x \text { is a nonsquare in } \mathbb{F}_q^*\end{cases}$ ,Let $\eta$ be the quadratic character of $\mathbb{F}_q^*$ , where $q=p^n$ , with $p$ an odd prime and $n \in \mathbb{Z}^{+}$ . Prove that

g(\eta)=(-1)^{(q+1) / 2} i^{n\left(p^2+2 p+5\right) / 4} q^{1 / 2} .

3.9 For $\chi_1, \ldots, \chi_k \in \widehat{\mathbb{F}_q^*}$ , define

J_0\left(\chi_1, \ldots, \chi_k\right)=\sum_{\substack{x_1, \ldots, x_k \in \mathbb{F}_q \\ x_1+\cdots+x_k=0}} \chi_1\left(x_1\right) \cdots \chi_k\left(x_k\right)

Prove that

\begin{aligned} & J_0\left(\chi_1, \ldots, \chi_k\right) \\ & = \begin{cases}q^{k-1} & \text { if } \chi_1=\cdots=\chi_k=1^{\mathbb{F}_q^*} \\ \frac{q-1}{q} g\left(\chi_1\right) \cdots g\left(\chi_k\right) & \text { if } \chi_i \neq 1^{\mathbb{F}_q^*}, 1 \leq i \leq k, \text { and } \chi_1 \cdots \chi_k=1^{\mathbb{F}_q^*} \\ 0 & \text { otherwise. }\end{cases} \end{aligned}

3.11 Prove that $f \in \mathbb{F}_q[\mathrm{X}]$ is a permutation polynomial of $\mathbb{F}_q$ if and only if $\sum_{x \in \mathbb{F}_q} e_a(f(x))=0 \quad$ for all $a \in \mathbb{F}_q^*$ .

3.13. Let $A$ and $B$ be finite abelian groups. A function $f: A \rightarrow B$ is said to be bent if for all $\alpha \in \widehat{A}$ and $\beta \in \widehat{B} \backslash\left\{1^B\right\}$ ,

\left|\sum_{x \in A} \alpha(x) \beta(f(x))\right|=|A|^{1 / 2} .

Prove that $f: A \rightarrow B$ is bent if and only if for each $x \in A \backslash\{0\}$ , the function $f_x: A \rightarrow B, f_x(y)=f(y+x)-f(y)$ , takes each value of $B$ an equal number of times.

Exercise: Assume that $y \in \mathbb{F}_{q^8}$ such that $y^{2\left(q^2-1\right)}=-1$ . Set $\chi(x)=$ $\zeta_p^{\mathrm{Tr}_{q^8 / p}(x)}$ where $x \in \mathbb{F}_{q^8}$ and $p=\operatorname{char}\left(\mathbb{F}_q\right)$ . Prove:

a) $\quad \sum_{a \in \mathbb{F}_{q^4}} \chi(a y)=q^4$ ;

b) There is no $z \in \mathbb{F}_{q^4}^*$ such that $y=b^{q^4}-b$ where $b=\eta z$ and $\eta$ is a non-square element in $\mathbb{F}_{q^8}$ .