Mixing
Question about rational points on elliptics curves.
$begingroup$
I have the following question about this lemma,
Where $A,Binmathbb{F}$. I tried to prove that $mbox{Norm}_{R/F}(alpha-betatheta)=f^{hom}(alpha,beta))$, but I don't see, why this is correct. Any hint will be appreciated. Just to be certain, in this case $mbox{Norm}_{R/mathbb{F}}$ is the norm in the finite field and $mbox{hom}$ is the homomorphism?
Thanks!
This is from Andrew Shallue's papers https://pdfs.semanticscholar.org/0139/0ce5986e93cc6ffa35050566760c96640c1a.pdf
elliptic-curves elliptic-equations
asked Jan 9 at 20:52
pablocn_pablocn_
371214
$endgroup$
add a comment |
$begingroup$
I have the following question about this lemma,
Where $A,Binmathbb{F}$. I tried to prove that $mbox{Norm}_{R/F}(alpha-betatheta)=f^{hom}(alpha,beta))$, but I don't see, why this is correct. Any hint will be appreciated. Just to be certain, in this case $mbox{Norm}_{R/mathbb{F}}$ is the norm in the finite field and $mbox{hom}$ is the homomorphism?
Thanks!
This is from Andrew Shallue's papers https://pdfs.semanticscholar.org/0139/0ce5986e93cc6ffa35050566760c96640c1a.pdf
elliptic-curves elliptic-equations
asked Jan 9 at 20:52
pablocn_pablocn_
371214
$endgroup$
$begingroup$
If $f in F[x] $ is monic irreducible separable with roots $theta_j$ and $R = F[x]/(f)$ and $alpha, beta in F$ then $Norm_{R/F}(alpha-theta_j beta) overset{def}= prod_{l=1}^{deg(f)} (alpha-theta_l beta) = beta^{deg(f)} f(alpha/beta)$. To show this is also $det(z mapsto (alpha-theta_j beta) z)$ you'd need some Galois theory. If $f = prod_m f_m$ is not irreducible but doesn't have double roots then the same holds because $R cong prod_m F[x]/(f_m)$.
$endgroup$
– reuns
Jan 9 at 21:43
add a comment |
$begingroup$
I have the following question about this lemma,
Where $A,Binmathbb{F}$. I tried to prove that $mbox{Norm}_{R/F}(alpha-betatheta)=f^{hom}(alpha,beta))$, but I don't see, why this is correct. Any hint will be appreciated. Just to be certain, in this case $mbox{Norm}_{R/mathbb{F}}$ is the norm in the finite field and $mbox{hom}$ is the homomorphism?
Thanks!
This is from Andrew Shallue's papers https://pdfs.semanticscholar.org/0139/0ce5986e93cc6ffa35050566760c96640c1a.pdf
elliptic-curves elliptic-equations
asked Jan 9 at 20:52
pablocn_pablocn_
371214
$endgroup$
I have the following question about this lemma,
Where $A,Binmathbb{F}$. I tried to prove that $mbox{Norm}_{R/F}(alpha-betatheta)=f^{hom}(alpha,beta))$, but I don't see, why this is correct. Any hint will be appreciated. Just to be certain, in this case $mbox{Norm}_{R/mathbb{F}}$ is the norm in the finite field and $mbox{hom}$ is the homomorphism?
Thanks!
This is from Andrew Shallue's papers https://pdfs.semanticscholar.org/0139/0ce5986e93cc6ffa35050566760c96640c1a.pdf
elliptic-curves elliptic-equations
elliptic-curves elliptic-equations
asked Jan 9 at 20:52
pablocn_pablocn_
371214
asked Jan 9 at 20:52
pablocn_pablocn_
371214
edited Jan 10 at 4:46
pablocn_
asked Jan 9 at 20:52
pablocn_pablocn_
371214
asked Jan 9 at 20:52
pablocn_pablocn_
371214
asked Jan 9 at 20:52
pablocn_pablocn_
371214
371214
$begingroup$
If $f in F[x] $ is monic irreducible separable with roots $theta_j$ and $R = F[x]/(f)$ and $alpha, beta in F$ then $Norm_{R/F}(alpha-theta_j beta) overset{def}= prod_{l=1}^{deg(f)} (alpha-theta_l beta) = beta^{deg(f)} f(alpha/beta)$. To show this is also $det(z mapsto (alpha-theta_j beta) z)$ you'd need some Galois theory. If $f = prod_m f_m$ is not irreducible but doesn't have double roots then the same holds because $R cong prod_m F[x]/(f_m)$.
$endgroup$
– reuns
Jan 9 at 21:43
add a comment |
$begingroup$
If $f in F[x] $ is monic irreducible separable with roots $theta_j$ and $R = F[x]/(f)$ and $alpha, beta in F$ then $Norm_{R/F}(alpha-theta_j beta) overset{def}= prod_{l=1}^{deg(f)} (alpha-theta_l beta) = beta^{deg(f)} f(alpha/beta)$. To show this is also $det(z mapsto (alpha-theta_j beta) z)$ you'd need some Galois theory. If $f = prod_m f_m$ is not irreducible but doesn't have double roots then the same holds because $R cong prod_m F[x]/(f_m)$.
$endgroup$
– reuns
Jan 9 at 21:43
$begingroup$
If $f in F[x] $ is monic irreducible separable with roots $theta_j$ and $R = F[x]/(f)$ and $alpha, beta in F$ then $Norm_{R/F}(alpha-theta_j beta) overset{def}= prod_{l=1}^{deg(f)} (alpha-theta_l beta) = beta^{deg(f)} f(alpha/beta)$. To show this is also $det(z mapsto (alpha-theta_j beta) z)$ you'd need some Galois theory. If $f = prod_m f_m$ is not irreducible but doesn't have double roots then the same holds because $R cong prod_m F[x]/(f_m)$.
$endgroup$
– reuns
Jan 9 at 21:43
$begingroup$
If $f in F[x] $ is monic irreducible separable with roots $theta_j$ and $R = F[x]/(f)$ and $alpha, beta in F$ then $Norm_{R/F}(alpha-theta_j beta) overset{def}= prod_{l=1}^{deg(f)} (alpha-theta_l beta) = beta^{deg(f)} f(alpha/beta)$. To show this is also $det(z mapsto (alpha-theta_j beta) z)$ you'd need some Galois theory. If $f = prod_m f_m$ is not irreducible but doesn't have double roots then the same holds because $R cong prod_m F[x]/(f_m)$.
$endgroup$
– reuns
Jan 9 at 21:43
add a comment |
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$begingroup$
If $f in F[x] $ is monic irreducible separable with roots $theta_j$ and $R = F[x]/(f)$ and $alpha, beta in F$ then $Norm_{R/F}(alpha-theta_j beta) overset{def}= prod_{l=1}^{deg(f)} (alpha-theta_l beta) = beta^{deg(f)} f(alpha/beta)$. To show this is also $det(z mapsto (alpha-theta_j beta) z)$ you'd need some Galois theory. If $f = prod_m f_m$ is not irreducible but doesn't have double roots then the same holds because $R cong prod_m F[x]/(f_m)$.
$endgroup$
– reuns
Jan 9 at 21:43