Jacobi sums Gaussian Sum. Show $J(rho^{'},chi^{'}) = (- J(rho,chi))^s$
$begingroup$
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?
number-theory characters gauss-sums
asked Jan 16 at 23:38
MemoriesMemories
11311
$endgroup$
add a comment |
$begingroup$
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?
number-theory characters gauss-sums
asked Jan 16 at 23:38
MemoriesMemories
11311
$endgroup$
add a comment |
$begingroup$
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?
number-theory characters gauss-sums
asked Jan 16 at 23:38
MemoriesMemories
11311
$endgroup$
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
number-theory characters gauss-sums
asked Jan 16 at 23:38
MemoriesMemories
11311
asked Jan 16 at 23:38
MemoriesMemories
11311
asked Jan 16 at 23:38
MemoriesMemories
11311
asked Jan 16 at 23:38
MemoriesMemories
11311
asked Jan 16 at 23:38
MemoriesMemories
11311
11311
add a comment |
add a comment |
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