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












1












$begingroup$


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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$endgroup$

















    1












    $begingroup$


    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?










    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      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?










      share|cite|improve this question









      $endgroup$




      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?







      number-theory characters gauss-sums






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      share|cite|improve this question










      asked Jan 16 at 23:38









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