Are all finite extensions of perfect fields cyclic?
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
|
show 1 more comment
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
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.
– user623904
Jan 4 at 7:40
There are noncyclic extensions of fields of characteristic zero.
– Lord Shark the Unknown
Jan 4 at 7:47
Not to mention that your link does not solve the problem, since not every perfect field of positive characteristic is finite...
– Kenny Lau
Jan 4 at 7:55
@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.
– user623904
Jan 5 at 5:50
1
Try $Bbb Q(sqrt 2, sqrt 3) / Bbb Q$
– Watson
2 days ago
|
show 1 more comment
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
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
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.
– user623904
Jan 4 at 7:40
There are noncyclic extensions of fields of characteristic zero.
– Lord Shark the Unknown
Jan 4 at 7:47
Not to mention that your link does not solve the problem, since not every perfect field of positive characteristic is finite...
– Kenny Lau
Jan 4 at 7:55
@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.
– user623904
Jan 5 at 5:50
1
Try $Bbb Q(sqrt 2, sqrt 3) / Bbb Q$
– Watson
2 days ago
|
show 1 more comment
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.
– user623904
Jan 4 at 7:40
There are noncyclic extensions of fields of characteristic zero.
– Lord Shark the Unknown
Jan 4 at 7:47
Not to mention that your link does not solve the problem, since not every perfect field of positive characteristic is finite...
– Kenny Lau
Jan 4 at 7:55
@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.
– user623904
Jan 5 at 5:50
1
Try $Bbb Q(sqrt 2, sqrt 3) / Bbb Q$
– Watson
2 days ago
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.
– user623904
Jan 4 at 7:40
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.
– user623904
Jan 4 at 7:40
There are noncyclic extensions of fields of characteristic zero.
– Lord Shark the Unknown
Jan 4 at 7:47
There are noncyclic extensions of fields of characteristic zero.
– Lord Shark the Unknown
Jan 4 at 7:47
Not to mention that your link does not solve the problem, since not every perfect field of positive characteristic is finite...
– Kenny Lau
Jan 4 at 7:55
Not to mention that your link does not solve the problem, since not every perfect field of positive characteristic is finite...
– Kenny Lau
Jan 4 at 7:55
@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.
– user623904
Jan 5 at 5:50
@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.
– user623904
Jan 5 at 5:50
1
1
Try $Bbb Q(sqrt 2, sqrt 3) / Bbb Q$
– Watson
2 days ago
Try $Bbb Q(sqrt 2, sqrt 3) / Bbb Q$
– Watson
2 days ago
|
show 1 more comment
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3061400%2fare-all-finite-extensions-of-perfect-fields-cyclic%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3061400%2fare-all-finite-extensions-of-perfect-fields-cyclic%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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.
– user623904
Jan 4 at 7:40
There are noncyclic extensions of fields of characteristic zero.
– Lord Shark the Unknown
Jan 4 at 7:47
Not to mention that your link does not solve the problem, since not every perfect field of positive characteristic is finite...
– Kenny Lau
Jan 4 at 7:55
@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.
– user623904
Jan 5 at 5:50
1
Try $Bbb Q(sqrt 2, sqrt 3) / Bbb Q$
– Watson
2 days ago