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Kummer extension and subgroups of $F^*/F^*n$
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In Book Morandi,P Field and Galois Theory we have:
Proposition 11.10: let $K/F$ be an n-Kummer extension. Then there is
an injective group homomorphism $f: operatornamekum(K/F)rightarrow F^*/F^*n$ given by $f(alpha F^*) = alpha^nF^*n$. The image of $f$ is then a finite subgroup of $F^*/F^*n$ of order equal to $[K:F]$.
($ operatornameKUM(K/F) = leftlbrace ain K^* : a^n in Frightrbrace$ and $ operatornamekum(K/F) = operatornameKUM(K/F)/F^*$)
Problem 11.6: Let $F$ be a field containing a primitive nth root of unity, and let $G$ be a subgroup of $F^*/F^*n$. Let $F(G) = F(leftlbrace sqrt[n]a : aF^*n in Grightrbrace)$. Show that $F(G)$ is an n-Kummer extension of $F$ and that $G$ is the image of $ operatornamekum(F(G)/F)$ under the map $f$ defined in Proposition 11.10. Conclude that $ operatornameGal(K/F)simeq G$ and $[F(G) : F] = |G|$.
My question is with these two fact can we say n-kummer extensions of $F$ is one-to-one correspondence with the subgroups of $F^*/F^*n$?
galois-theory kummer-theory
asked Jul 24 at 15:05
s.Bahari
384
add a comment |Â
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0
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In Book Morandi,P Field and Galois Theory we have:
Proposition 11.10: let $K/F$ be an n-Kummer extension. Then there is
an injective group homomorphism $f: operatornamekum(K/F)rightarrow F^*/F^*n$ given by $f(alpha F^*) = alpha^nF^*n$. The image of $f$ is then a finite subgroup of $F^*/F^*n$ of order equal to $[K:F]$.
($ operatornameKUM(K/F) = leftlbrace ain K^* : a^n in Frightrbrace$ and $ operatornamekum(K/F) = operatornameKUM(K/F)/F^*$)
Problem 11.6: Let $F$ be a field containing a primitive nth root of unity, and let $G$ be a subgroup of $F^*/F^*n$. Let $F(G) = F(leftlbrace sqrt[n]a : aF^*n in Grightrbrace)$. Show that $F(G)$ is an n-Kummer extension of $F$ and that $G$ is the image of $ operatornamekum(F(G)/F)$ under the map $f$ defined in Proposition 11.10. Conclude that $ operatornameGal(K/F)simeq G$ and $[F(G) : F] = |G|$.
My question is with these two fact can we say n-kummer extensions of $F$ is one-to-one correspondence with the subgroups of $F^*/F^*n$?
galois-theory kummer-theory
asked Jul 24 at 15:05
s.Bahari
384
Yes, but Kummer's theorem is more precise than just a bijection, see e.g. math.stackexchange.com/a/2539569/300700
– nguyen quang do
Jul 28 at 6:44
add a comment |Â
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up vote
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down vote
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In Book Morandi,P Field and Galois Theory we have:
Proposition 11.10: let $K/F$ be an n-Kummer extension. Then there is
an injective group homomorphism $f: operatornamekum(K/F)rightarrow F^*/F^*n$ given by $f(alpha F^*) = alpha^nF^*n$. The image of $f$ is then a finite subgroup of $F^*/F^*n$ of order equal to $[K:F]$.
($ operatornameKUM(K/F) = leftlbrace ain K^* : a^n in Frightrbrace$ and $ operatornamekum(K/F) = operatornameKUM(K/F)/F^*$)
Problem 11.6: Let $F$ be a field containing a primitive nth root of unity, and let $G$ be a subgroup of $F^*/F^*n$. Let $F(G) = F(leftlbrace sqrt[n]a : aF^*n in Grightrbrace)$. Show that $F(G)$ is an n-Kummer extension of $F$ and that $G$ is the image of $ operatornamekum(F(G)/F)$ under the map $f$ defined in Proposition 11.10. Conclude that $ operatornameGal(K/F)simeq G$ and $[F(G) : F] = |G|$.
My question is with these two fact can we say n-kummer extensions of $F$ is one-to-one correspondence with the subgroups of $F^*/F^*n$?
galois-theory kummer-theory
asked Jul 24 at 15:05
s.Bahari
384
In Book Morandi,P Field and Galois Theory we have:
Proposition 11.10: let $K/F$ be an n-Kummer extension. Then there is
an injective group homomorphism $f: operatornamekum(K/F)rightarrow F^*/F^*n$ given by $f(alpha F^*) = alpha^nF^*n$. The image of $f$ is then a finite subgroup of $F^*/F^*n$ of order equal to $[K:F]$.
($ operatornameKUM(K/F) = leftlbrace ain K^* : a^n in Frightrbrace$ and $ operatornamekum(K/F) = operatornameKUM(K/F)/F^*$)
Problem 11.6: Let $F$ be a field containing a primitive nth root of unity, and let $G$ be a subgroup of $F^*/F^*n$. Let $F(G) = F(leftlbrace sqrt[n]a : aF^*n in Grightrbrace)$. Show that $F(G)$ is an n-Kummer extension of $F$ and that $G$ is the image of $ operatornamekum(F(G)/F)$ under the map $f$ defined in Proposition 11.10. Conclude that $ operatornameGal(K/F)simeq G$ and $[F(G) : F] = |G|$.
My question is with these two fact can we say n-kummer extensions of $F$ is one-to-one correspondence with the subgroups of $F^*/F^*n$?
galois-theory kummer-theory
asked Jul 24 at 15:05
s.Bahari
384
edited Jul 25 at 15:17
asked Jul 24 at 15:05
s.Bahari
384
asked Jul 24 at 15:05
s.Bahari
384
asked Jul 24 at 15:05
s.Bahari
384
384
Yes, but Kummer's theorem is more precise than just a bijection, see e.g. math.stackexchange.com/a/2539569/300700
– nguyen quang do
Jul 28 at 6:44
add a comment |Â
Yes, but Kummer's theorem is more precise than just a bijection, see e.g. math.stackexchange.com/a/2539569/300700
– nguyen quang do
Jul 28 at 6:44
Yes, but Kummer's theorem is more precise than just a bijection, see e.g. math.stackexchange.com/a/2539569/300700
– nguyen quang do
Jul 28 at 6:44
Yes, but Kummer's theorem is more precise than just a bijection, see e.g. math.stackexchange.com/a/2539569/300700
– nguyen quang do
Jul 28 at 6:44
add a comment |Â
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Yes, but Kummer's theorem is more precise than just a bijection, see e.g. math.stackexchange.com/a/2539569/300700
– nguyen quang do
Jul 28 at 6:44