Are all finite extensions of perfect fields cyclic?
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I am not well trained in number theory. If there is any mistake, please straightly edit this question or devote me if the question is too trivial.
According to https://en.wikipedia.org/wiki/Perfect_field,
a field $k$ is perfect iff $char(k)=0$ or $char(k)=p>0$ with Frobenius endomorphism being an isomorphism of $k$.
I think at least the Pete L. Clark's answer in Are all extensions of finite fields cyclic? having gently solved the $char(k)=p>0$ case of the problem, but the same method can not be applied to characteristic zero cases.
algebraic-number-theory extension-field
asked Jan 4 at 7:39
user623904user623904
234
$endgroup$
|
show 1 more comment
$begingroup$
I am not well trained in number theory. If there is any mistake, please straightly edit this question or devote me if the question is too trivial.
According to https://en.wikipedia.org/wiki/Perfect_field,
a field $k$ is perfect iff $char(k)=0$ or $char(k)=p>0$ with Frobenius endomorphism being an isomorphism of $k$.
I think at least the Pete L. Clark's answer in Are all extensions of finite fields cyclic? having gently solved the $char(k)=p>0$ case of the problem, but the same method can not be applied to characteristic zero cases.
algebraic-number-theory extension-field
asked Jan 4 at 7:39
user623904user623904
234
$endgroup$
$begingroup$
Addendum: This is a fact mentioned in Serre's $it{Local,, Field}$, giving the motivation for a claim. Let $k_{n}$ be a cyclic extension of $k$ of degree $n$ and $F^{n}$ be the generator of $G_{k_{n}/k}$. This claim is, if $k$ is perfect and $k^{sep}$ is the union of all cyclic extensions $k^{n}$ for all $n$ with $F^{mn}|_{k_{n}}=F^{n}$, then $k$ is a quasi-finite field.
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– user623904
Jan 4 at 7:40
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There are noncyclic extensions of fields of characteristic zero.
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– Lord Shark the Unknown
Jan 4 at 7:47
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Not to mention that your link does not solve the problem, since not every perfect field of positive characteristic is finite...
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– Kenny Lau
Jan 4 at 7:55
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@Lord Shark the Unknown I will grateful if you can provide any explicit example for a finite extension of a characteristic zero field with Galois group which is not cyclic, and I think a counterexample is also an answer for this problem. I failed to find an example at present.
$endgroup$
– user623904
Jan 5 at 5:50
1
$begingroup$
Try $Bbb Q(sqrt 2, sqrt 3) / Bbb Q$
$endgroup$
– Watson
Jan 5 at 9:47
|
show 1 more comment
$begingroup$
I am not well trained in number theory. If there is any mistake, please straightly edit this question or devote me if the question is too trivial.
According to https://en.wikipedia.org/wiki/Perfect_field,
a field $k$ is perfect iff $char(k)=0$ or $char(k)=p>0$ with Frobenius endomorphism being an isomorphism of $k$.
I think at least the Pete L. Clark's answer in Are all extensions of finite fields cyclic? having gently solved the $char(k)=p>0$ case of the problem, but the same method can not be applied to characteristic zero cases.
algebraic-number-theory extension-field
asked Jan 4 at 7:39
user623904user623904
234
$endgroup$
I am not well trained in number theory. If there is any mistake, please straightly edit this question or devote me if the question is too trivial.
According to https://en.wikipedia.org/wiki/Perfect_field,
a field $k$ is perfect iff $char(k)=0$ or $char(k)=p>0$ with Frobenius endomorphism being an isomorphism of $k$.
I think at least the Pete L. Clark's answer in Are all extensions of finite fields cyclic? having gently solved the $char(k)=p>0$ case of the problem, but the same method can not be applied to characteristic zero cases.
algebraic-number-theory extension-field
algebraic-number-theory extension-field
asked Jan 4 at 7:39
user623904user623904
234
asked Jan 4 at 7:39
user623904user623904
234
edited Jan 4 at 8:17
user623904
asked Jan 4 at 7:39
user623904user623904
234
asked Jan 4 at 7:39
user623904user623904
234
asked Jan 4 at 7:39
user623904user623904
234
234
$begingroup$
Addendum: This is a fact mentioned in Serre's $it{Local,, Field}$, giving the motivation for a claim. Let $k_{n}$ be a cyclic extension of $k$ of degree $n$ and $F^{n}$ be the generator of $G_{k_{n}/k}$. This claim is, if $k$ is perfect and $k^{sep}$ is the union of all cyclic extensions $k^{n}$ for all $n$ with $F^{mn}|_{k_{n}}=F^{n}$, then $k$ is a quasi-finite field.
$endgroup$
– user623904
Jan 4 at 7:40
$begingroup$
There are noncyclic extensions of fields of characteristic zero.
$endgroup$
– Lord Shark the Unknown
Jan 4 at 7:47
$begingroup$
Not to mention that your link does not solve the problem, since not every perfect field of positive characteristic is finite...
$endgroup$
– Kenny Lau
Jan 4 at 7:55
$begingroup$
@Lord Shark the Unknown I will grateful if you can provide any explicit example for a finite extension of a characteristic zero field with Galois group which is not cyclic, and I think a counterexample is also an answer for this problem. I failed to find an example at present.
$endgroup$
– user623904
Jan 5 at 5:50
1
$begingroup$
Try $Bbb Q(sqrt 2, sqrt 3) / Bbb Q$
$endgroup$
– Watson
Jan 5 at 9:47
|
show 1 more comment
$begingroup$
Addendum: This is a fact mentioned in Serre's $it{Local,, Field}$, giving the motivation for a claim. Let $k_{n}$ be a cyclic extension of $k$ of degree $n$ and $F^{n}$ be the generator of $G_{k_{n}/k}$. This claim is, if $k$ is perfect and $k^{sep}$ is the union of all cyclic extensions $k^{n}$ for all $n$ with $F^{mn}|_{k_{n}}=F^{n}$, then $k$ is a quasi-finite field.
$endgroup$
– user623904
Jan 4 at 7:40
$begingroup$
There are noncyclic extensions of fields of characteristic zero.
$endgroup$
– Lord Shark the Unknown
Jan 4 at 7:47
$begingroup$
Not to mention that your link does not solve the problem, since not every perfect field of positive characteristic is finite...
$endgroup$
– Kenny Lau
Jan 4 at 7:55
$begingroup$
@Lord Shark the Unknown I will grateful if you can provide any explicit example for a finite extension of a characteristic zero field with Galois group which is not cyclic, and I think a counterexample is also an answer for this problem. I failed to find an example at present.
$endgroup$
– user623904
Jan 5 at 5:50
1
$begingroup$
Try $Bbb Q(sqrt 2, sqrt 3) / Bbb Q$
$endgroup$
– Watson
Jan 5 at 9:47
$begingroup$
Addendum: This is a fact mentioned in Serre's $it{Local,, Field}$, giving the motivation for a claim. Let $k_{n}$ be a cyclic extension of $k$ of degree $n$ and $F^{n}$ be the generator of $G_{k_{n}/k}$. This claim is, if $k$ is perfect and $k^{sep}$ is the union of all cyclic extensions $k^{n}$ for all $n$ with $F^{mn}|_{k_{n}}=F^{n}$, then $k$ is a quasi-finite field.
$endgroup$
– user623904
Jan 4 at 7:40
$begingroup$
Addendum: This is a fact mentioned in Serre's $it{Local,, Field}$, giving the motivation for a claim. Let $k_{n}$ be a cyclic extension of $k$ of degree $n$ and $F^{n}$ be the generator of $G_{k_{n}/k}$. This claim is, if $k$ is perfect and $k^{sep}$ is the union of all cyclic extensions $k^{n}$ for all $n$ with $F^{mn}|_{k_{n}}=F^{n}$, then $k$ is a quasi-finite field.
$endgroup$
– user623904
Jan 4 at 7:40
$begingroup$
There are noncyclic extensions of fields of characteristic zero.
$endgroup$
– Lord Shark the Unknown
Jan 4 at 7:47
$begingroup$
There are noncyclic extensions of fields of characteristic zero.
$endgroup$
– Lord Shark the Unknown
Jan 4 at 7:47
$begingroup$
Not to mention that your link does not solve the problem, since not every perfect field of positive characteristic is finite...
$endgroup$
– Kenny Lau
Jan 4 at 7:55
$begingroup$
Not to mention that your link does not solve the problem, since not every perfect field of positive characteristic is finite...
$endgroup$
– Kenny Lau
Jan 4 at 7:55
$begingroup$
@Lord Shark the Unknown I will grateful if you can provide any explicit example for a finite extension of a characteristic zero field with Galois group which is not cyclic, and I think a counterexample is also an answer for this problem. I failed to find an example at present.
$endgroup$
– user623904
Jan 5 at 5:50
$begingroup$
@Lord Shark the Unknown I will grateful if you can provide any explicit example for a finite extension of a characteristic zero field with Galois group which is not cyclic, and I think a counterexample is also an answer for this problem. I failed to find an example at present.
$endgroup$
– user623904
Jan 5 at 5:50
1
1
$begingroup$
Try $Bbb Q(sqrt 2, sqrt 3) / Bbb Q$
$endgroup$
– Watson
Jan 5 at 9:47
$begingroup$
Try $Bbb Q(sqrt 2, sqrt 3) / Bbb Q$
$endgroup$
– Watson
Jan 5 at 9:47
|
show 1 more comment
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$begingroup$
Addendum: This is a fact mentioned in Serre's $it{Local,, Field}$, giving the motivation for a claim. Let $k_{n}$ be a cyclic extension of $k$ of degree $n$ and $F^{n}$ be the generator of $G_{k_{n}/k}$. This claim is, if $k$ is perfect and $k^{sep}$ is the union of all cyclic extensions $k^{n}$ for all $n$ with $F^{mn}|_{k_{n}}=F^{n}$, then $k$ is a quasi-finite field.
$endgroup$
– user623904
Jan 4 at 7:40
$begingroup$
There are noncyclic extensions of fields of characteristic zero.
$endgroup$
– Lord Shark the Unknown
Jan 4 at 7:47
$begingroup$
Not to mention that your link does not solve the problem, since not every perfect field of positive characteristic is finite...
$endgroup$
– Kenny Lau
Jan 4 at 7:55
$begingroup$
@Lord Shark the Unknown I will grateful if you can provide any explicit example for a finite extension of a characteristic zero field with Galois group which is not cyclic, and I think a counterexample is also an answer for this problem. I failed to find an example at present.
$endgroup$
– user623904
Jan 5 at 5:50
1
$begingroup$
Try $Bbb Q(sqrt 2, sqrt 3) / Bbb Q$
$endgroup$
– Watson
Jan 5 at 9:47