# Computing number of solutions for equations in $F_{m^s}$ (finite field with $m^s$ elements)

First I want to give you some context. Then I will ask my questions.

Context

Consider the equations $$a_1x_1^{l_1} + \dots + a_rx_r^{l_r} = b$$ with $$a_1, \dots , a_r \in F_m^{*}$$, where $$F_m$$ is a finite field with $$m$$ elements. $$N$$ is the number of solutions. Now I know that if $$b \neq 0$$ :

$$N = m^{r-1} + \sum \chi_1\chi_2\cdots\chi_r(b) \chi_1(a_1^{-1})\chi_2(a_2^{-1})\cdots\chi_r(a_r^{-1})J(\chi_1,\dots,\chi_r)$$. The summation is over $$r$$-tuples of characters $$\chi_1, \dots, \chi_r$$, where $$\chi_i^{l_i} = \varepsilon$$ and $$\chi_i \neq \epsilon$$ for $$i = 1, \dots, r$$.

I want to specializing this to $$x^2 + y^4 = 1$$. Obviously $$r=2$$. So I want to sum over all $$2$$-tuples of characters $$\chi_1, \chi_2$$ where $$\chi_i^{l_i} = \varepsilon$$ and $$\chi_i \neq \epsilon$$ for $$i = 1,2$$.

So $$N = m + J(\rho,\chi) + J(\rho,\chi^2) + J(\rho,\chi^3)$$ $$= m - 1 + J(\rho,\chi) + J(\rho,\chi^3) = m - 1 + J(\rho,\chi) + \overline{J(\rho,\chi)}$$ where $$\rho$$ is a character of order $$2$$ and $$\chi$$ is a character of order $$4$$.

First Question ( Answered, I won ) :

I wanted to use that fact above again, but this time for $$F_{m^s}$$. At this point I had a discussion with my teacher. He said that the fact above is only for $$F_m$$, but I don't see a problem because I could say $$\hat{m} := m^s$$ or do you see a problem?

Second Question:

The number of solutions in $$F_{m^s}$$ has to be $$N = m^s - 1 - (-J(\rho,\chi))^s - (- J(\overline{\rho,\chi}))^s$$. But to be honest: I don't see why. I only know that we have to work with the Hasse-Davenport Relation. So we have to use the equation: $$(-g(\chi))^s = -g(\chi^{'})$$ (gaussian sums), where $$\chi$$ is a character of $$F_m$$ and $$\chi^{'}$$ is a character of $$F_{m^s}$$. I think that it won't be necessary to give you a exact form of $$\chi^{'}$$.