Question about fixed field of subgroup of Galois group.












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$ 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) $?










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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


















0












$begingroup$



$ 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) $?










share|cite|improve this question









$endgroup$








  • 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
















0












0








0


1



$begingroup$



$ 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) $?










share|cite|improve this question









$endgroup$





$ 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






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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
















  • 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












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