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?


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