Upper bound for class number of field
Suppose $K$ is a number field. Let $p$ be the smallest prime which is the norm of some principal ideal. It follows that every class in the ideal class group of $K$ contains some ideal of norm prime $q<p$.
Example of conjecture above:
The class number of the field $K=sqrt-271$ is $11$ and the smallest prime $p$ which is a norm of some principal ideal in $K$ is $307$. This means that every other class in the ideal class group of $K$ is generated by an ideal of norm $q$ NOT exceeding $307$. (Compare this to the Minkowski's bound).
elementary-number-theory class-field-theory
asked Dec 4 '18 at 2:04
J. Linne
846315
add a comment |
Suppose $K$ is a number field. Let $p$ be the smallest prime which is the norm of some principal ideal. It follows that every class in the ideal class group of $K$ contains some ideal of norm prime $q<p$.
Example of conjecture above:
The class number of the field $K=sqrt-271$ is $11$ and the smallest prime $p$ which is a norm of some principal ideal in $K$ is $307$. This means that every other class in the ideal class group of $K$ is generated by an ideal of norm $q$ NOT exceeding $307$. (Compare this to the Minkowski's bound).
elementary-number-theory class-field-theory
asked Dec 4 '18 at 2:04
J. Linne
846315
add a comment |
Suppose $K$ is a number field. Let $p$ be the smallest prime which is the norm of some principal ideal. It follows that every class in the ideal class group of $K$ contains some ideal of norm prime $q<p$.
Example of conjecture above:
The class number of the field $K=sqrt-271$ is $11$ and the smallest prime $p$ which is a norm of some principal ideal in $K$ is $307$. This means that every other class in the ideal class group of $K$ is generated by an ideal of norm $q$ NOT exceeding $307$. (Compare this to the Minkowski's bound).
elementary-number-theory class-field-theory
asked Dec 4 '18 at 2:04
J. Linne
846315
Suppose $K$ is a number field. Let $p$ be the smallest prime which is the norm of some principal ideal. It follows that every class in the ideal class group of $K$ contains some ideal of norm prime $q<p$.
Example of conjecture above:
The class number of the field $K=sqrt-271$ is $11$ and the smallest prime $p$ which is a norm of some principal ideal in $K$ is $307$. This means that every other class in the ideal class group of $K$ is generated by an ideal of norm $q$ NOT exceeding $307$. (Compare this to the Minkowski's bound).
elementary-number-theory class-field-theory
elementary-number-theory class-field-theory
asked Dec 4 '18 at 2:04
J. Linne
846315
asked Dec 4 '18 at 2:04
J. Linne
846315
asked Dec 4 '18 at 2:04
J. Linne
846315
asked Dec 4 '18 at 2:04
J. Linne
846315
asked Dec 4 '18 at 2:04
J. Linne
846315
846315
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
Every number field with class number $> 1$ in which $2$ splits completely and the prime ideals above $2$ are principal is a counterexample. For a start, look at quadratic number fields generated by $sqrt{t^2-2}$; here $t + sqrt{t^2 - 2}$ has norm $2$. The first examples are ${mathbb Q}(sqrt{34})$ and ${mathbb Q}(sqrt{79})$.
answered Dec 27 '18 at 18:46
franz lemmermeyer
6,98021847
add a comment |
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%2f3025028%2fupper-bound-for-class-number-of-field%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
Every number field with class number $> 1$ in which $2$ splits completely and the prime ideals above $2$ are principal is a counterexample. For a start, look at quadratic number fields generated by $sqrt{t^2-2}$; here $t + sqrt{t^2 - 2}$ has norm $2$. The first examples are ${mathbb Q}(sqrt{34})$ and ${mathbb Q}(sqrt{79})$.
answered Dec 27 '18 at 18:46
franz lemmermeyer
6,98021847
add a comment |
Every number field with class number $> 1$ in which $2$ splits completely and the prime ideals above $2$ are principal is a counterexample. For a start, look at quadratic number fields generated by $sqrt{t^2-2}$; here $t + sqrt{t^2 - 2}$ has norm $2$. The first examples are ${mathbb Q}(sqrt{34})$ and ${mathbb Q}(sqrt{79})$.
answered Dec 27 '18 at 18:46
franz lemmermeyer
6,98021847
add a comment |
Every number field with class number $> 1$ in which $2$ splits completely and the prime ideals above $2$ are principal is a counterexample. For a start, look at quadratic number fields generated by $sqrt{t^2-2}$; here $t + sqrt{t^2 - 2}$ has norm $2$. The first examples are ${mathbb Q}(sqrt{34})$ and ${mathbb Q}(sqrt{79})$.
answered Dec 27 '18 at 18:46
franz lemmermeyer
6,98021847
Every number field with class number $> 1$ in which $2$ splits completely and the prime ideals above $2$ are principal is a counterexample. For a start, look at quadratic number fields generated by $sqrt{t^2-2}$; here $t + sqrt{t^2 - 2}$ has norm $2$. The first examples are ${mathbb Q}(sqrt{34})$ and ${mathbb Q}(sqrt{79})$.
answered Dec 27 '18 at 18:46
franz lemmermeyer
6,98021847
answered Dec 27 '18 at 18:46
franz lemmermeyer
6,98021847
answered Dec 27 '18 at 18:46
franz lemmermeyer
6,98021847
answered Dec 27 '18 at 18:46
franz lemmermeyer
6,98021847
6,98021847
add a comment |
add a comment |
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%2f3025028%2fupper-bound-for-class-number-of-field%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
