I’m trying to follow the book A Classical Introduction to Modern Number Theory byIreland and Rosen, but I have a couple of question about two proofs, so I will write down some parts of the proofs, showing where I have doubts, hope you can help me
A multiplicative character on $F_p$ is a map $\chi$ from $F_p^*$ to the nonzero complex numbers that satisfies $$\chi(ab)=\chi(a)\chi(b)$$ for all $a,b \in F_p^*$
Let $\chi, \lambda$ be characters of $F_p$ and set $J(\chi, \lambda) = \sum_{a+b=1} \chi (a)\lambda(b)$
First question (first part of section 4 page 98) Suppose we have the following, for $p\equiv 1 \pmod n$ $$N(x^n+y^n=1) = \sum_{j=0}^{n-1} \sum_{i=0}^{n-1}J(\chi^j,\chi^i)$$ so we want to estimate the sum. So, when $i=j=0$ we have $J(\chi^0,\chi^0)=J(\epsilon,\epsilon) =p$. When $j+i=n$, we have that $\chi^j=(\chi^j)^{-1}$(why?), so that $J(\chi^j,\chi^i) = -\chi^j(-1)$. The sum of these terms is $-\sum_{j=1}^{n-1} \chi^{j}(-1)$. $\sum_{j=0}^{n-1}\chi^j(-1)$ is $n$ when $-1$ is an $n$th power and zero otherwise. Thus the contribution of these terms is $1-\delta_n(-1)n$ where $d_n{-1}$ has te obvious meaning.(why the sum is $n$ when $-1$ is and $n$th power). Finally, if $i=0$ and $j\neq 0$ or $i\neq 0$ and $j=0$, then $J(\chi^i,\chi^j)=0$. Thus $$N(x^n+y^n=1) = p + 1-\delta_n(-1)n + \sum_{i,j}J(\chi^i,\chi^j)$$ (how do we get this last equality or expression?)
Second question(section 5 proposition 8.5.1 part (b) page 99) Assume that $\chi_1,\chi_2,\ldots,\chi_s$ are nontrivial and that $\chi_{s+1}=\chi_{s+2} = \cdots = \chi_l = \epsilon.$ Then $$\sum_{t_1+\cdots +t_l=0} \chi_1(t_1)\chi_2(t_2)\cdots \chi_l(t_l) = \sum_{t_1,t_2,\ldots,t_{l-1}} \chi_1(t_1)\chi_2(t_2)\cdots \chi_l(t_s)$$
(why is this equality or expression true?)
Thank you in advance