Question about fixed field of subgroup of Galois group.
$begingroup$
$ F=K(alpha) $ is a finite Galois extension of field $ K $. Let $ G $ be the Galois group, and $ H $ is a subgroup of $ G $. Define $ f(x)=prod_{sigmain H}(x-sigma(alpha)) $, prove that the fixed field of $ H $ is generated by $ K $ and all the coefficients of $ f(x) $.
Can someone give me some hints about how the fixed field of $ H $ is related to the coefficients of $ f(x) $?
abstract-algebra galois-theory
asked Dec 19 '18 at 6:53
user549397user549397
1,5081418
$endgroup$
add a comment |
$begingroup$
$ F=K(alpha) $ is a finite Galois extension of field $ K $. Let $ G $ be the Galois group, and $ H $ is a subgroup of $ G $. Define $ f(x)=prod_{sigmain H}(x-sigma(alpha)) $, prove that the fixed field of $ H $ is generated by $ K $ and all the coefficients of $ f(x) $.
Can someone give me some hints about how the fixed field of $ H $ is related to the coefficients of $ f(x) $?
abstract-algebra galois-theory
asked Dec 19 '18 at 6:53
user549397user549397
1,5081418
$endgroup$
1
$begingroup$
Can you show the coefficients of $f$ lie in the fixed field of $H$?
$endgroup$
– Lord Shark the Unknown
Dec 19 '18 at 7:09
$begingroup$
$f(x) = prod_{sigma in H} (x-sigma(alpha)) = sum_{n=0}^N a_n x^n$ then for any ring morphism $rho$, $sum_{n=0}^N rho(a_n) x^n = prod_{sigma in H} (x-rho(sigma(alpha)))$. What if $rho in H$ ? Then let $beta in F^H, beta = sum_j c_j alpha^j = sum_j c_j frac{1}{|H|}sum_{sigmain H}sigma(alpha)^j $
$endgroup$
– reuns
Dec 19 '18 at 10:57
add a comment |
$begingroup$
$ F=K(alpha) $ is a finite Galois extension of field $ K $. Let $ G $ be the Galois group, and $ H $ is a subgroup of $ G $. Define $ f(x)=prod_{sigmain H}(x-sigma(alpha)) $, prove that the fixed field of $ H $ is generated by $ K $ and all the coefficients of $ f(x) $.
Can someone give me some hints about how the fixed field of $ H $ is related to the coefficients of $ f(x) $?
abstract-algebra galois-theory
asked Dec 19 '18 at 6:53
user549397user549397
1,5081418
$endgroup$
$ F=K(alpha) $ is a finite Galois extension of field $ K $. Let $ G $ be the Galois group, and $ H $ is a subgroup of $ G $. Define $ f(x)=prod_{sigmain H}(x-sigma(alpha)) $, prove that the fixed field of $ H $ is generated by $ K $ and all the coefficients of $ f(x) $.
Can someone give me some hints about how the fixed field of $ H $ is related to the coefficients of $ f(x) $?
abstract-algebra galois-theory
abstract-algebra galois-theory
asked Dec 19 '18 at 6:53
user549397user549397
1,5081418
asked Dec 19 '18 at 6:53
user549397user549397
1,5081418
asked Dec 19 '18 at 6:53
user549397user549397
1,5081418
asked Dec 19 '18 at 6:53
user549397user549397
1,5081418
asked Dec 19 '18 at 6:53
user549397user549397
1,5081418
1,5081418
1
$begingroup$
Can you show the coefficients of $f$ lie in the fixed field of $H$?
$endgroup$
– Lord Shark the Unknown
Dec 19 '18 at 7:09
$begingroup$
$f(x) = prod_{sigma in H} (x-sigma(alpha)) = sum_{n=0}^N a_n x^n$ then for any ring morphism $rho$, $sum_{n=0}^N rho(a_n) x^n = prod_{sigma in H} (x-rho(sigma(alpha)))$. What if $rho in H$ ? Then let $beta in F^H, beta = sum_j c_j alpha^j = sum_j c_j frac{1}{|H|}sum_{sigmain H}sigma(alpha)^j $
$endgroup$
– reuns
Dec 19 '18 at 10:57
add a comment |
1
$begingroup$
Can you show the coefficients of $f$ lie in the fixed field of $H$?
$endgroup$
– Lord Shark the Unknown
Dec 19 '18 at 7:09
$begingroup$
$f(x) = prod_{sigma in H} (x-sigma(alpha)) = sum_{n=0}^N a_n x^n$ then for any ring morphism $rho$, $sum_{n=0}^N rho(a_n) x^n = prod_{sigma in H} (x-rho(sigma(alpha)))$. What if $rho in H$ ? Then let $beta in F^H, beta = sum_j c_j alpha^j = sum_j c_j frac{1}{|H|}sum_{sigmain H}sigma(alpha)^j $
$endgroup$
– reuns
Dec 19 '18 at 10:57
1
1
$begingroup$
Can you show the coefficients of $f$ lie in the fixed field of $H$?
$endgroup$
– Lord Shark the Unknown
Dec 19 '18 at 7:09
$begingroup$
Can you show the coefficients of $f$ lie in the fixed field of $H$?
$endgroup$
– Lord Shark the Unknown
Dec 19 '18 at 7:09
$begingroup$
$f(x) = prod_{sigma in H} (x-sigma(alpha)) = sum_{n=0}^N a_n x^n$ then for any ring morphism $rho$, $sum_{n=0}^N rho(a_n) x^n = prod_{sigma in H} (x-rho(sigma(alpha)))$. What if $rho in H$ ? Then let $beta in F^H, beta = sum_j c_j alpha^j = sum_j c_j frac{1}{|H|}sum_{sigmain H}sigma(alpha)^j $
$endgroup$
– reuns
Dec 19 '18 at 10:57
$begingroup$
$f(x) = prod_{sigma in H} (x-sigma(alpha)) = sum_{n=0}^N a_n x^n$ then for any ring morphism $rho$, $sum_{n=0}^N rho(a_n) x^n = prod_{sigma in H} (x-rho(sigma(alpha)))$. What if $rho in H$ ? Then let $beta in F^H, beta = sum_j c_j alpha^j = sum_j c_j frac{1}{|H|}sum_{sigmain H}sigma(alpha)^j $
$endgroup$
– reuns
Dec 19 '18 at 10:57
add a comment |
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1
$begingroup$
Can you show the coefficients of $f$ lie in the fixed field of $H$?
$endgroup$
– Lord Shark the Unknown
Dec 19 '18 at 7:09
$begingroup$
$f(x) = prod_{sigma in H} (x-sigma(alpha)) = sum_{n=0}^N a_n x^n$ then for any ring morphism $rho$, $sum_{n=0}^N rho(a_n) x^n = prod_{sigma in H} (x-rho(sigma(alpha)))$. What if $rho in H$ ? Then let $beta in F^H, beta = sum_j c_j alpha^j = sum_j c_j frac{1}{|H|}sum_{sigmain H}sigma(alpha)^j $
$endgroup$
– reuns
Dec 19 '18 at 10:57