# Jacobi sums Gaussian Sum. Show $J(\rho^{'},\chi^{'}) = (- J(\rho,\chi))^s$

I want to show that $$J(\rho^{'},\chi^{'}) = (- J(\rho,\chi))^s$$, where $$\rho^{'},\chi^{'}$$ are characters of a finite field $$F_{p^s}$$ and $$\chi,\rho$$ are characters a finite field $$F_p$$.

My work: I know / find out that

1) $$J(\rho^{'},\chi^{'}) = \frac{g(\chi^{'})g(\rho^{'})}{g(\rho^{'}\chi^{'})}$$

2) $$g(\chi^{'}) = (-g(\chi))^s$$

That should be enough to prove my claim, but how exactly? In 1) I can replace numerator with 2). But what now?