有限域作业

3.8 $\eta(x)= \begin{cases}1 & \text { if } x \text { is a square in } \mathbb{F}_q^* \ -1 & \text {

2.1. Let $f=\mathrm{X}^2+b \mathrm{X}+c \in \mathbb{F}_q[\mathrm{X}]$. (i) If $q$ is odd, prove that

2.3. Let $a, b$ be elements of $\mathbb{F}_{2^n}, n$ odd. Show that $a^2+a b+b^2=0$ implies $a=b=0$.

$Proof$:我们已经证明了$\mathbb{Z}_n$在加法$+_n$下为交换群，若想说明$\mathbb{Z}_n$构成交换环，还需验证结合律和乘法分配律 结合律：对于任意的 $a, b, c