Consider the order $mathbb{Z}[sqrt[4]{24}]$. Find all ideals of norm 100.
$begingroup$
I have found that the ring of integers is $mathbb{Z}[alpha, alpha^3/4]$ where $alpha = sqrt[4]{24}$. I also know that in the ring of integers $(5)$ factors as two ideals of norm $25$, and $(2)$ factors as $(2, alpha)^4$ hence there are 2 ideals of norm $100$. How does this help me find the ideals of norm $100$ in the smaller order (where there is no longer unique factorization in prime ideals)?
I know there still is a primary decomposition of any ideal but I don't see how to find all ideals of norm $8$ above $(2)$ in $mathbb{Z}[alpha]$ for example.
algebraic-number-theory norm ideals
asked Jan 16 at 17:02
RubiskRubisk
112
$endgroup$
add a comment |
$begingroup$
I have found that the ring of integers is $mathbb{Z}[alpha, alpha^3/4]$ where $alpha = sqrt[4]{24}$. I also know that in the ring of integers $(5)$ factors as two ideals of norm $25$, and $(2)$ factors as $(2, alpha)^4$ hence there are 2 ideals of norm $100$. How does this help me find the ideals of norm $100$ in the smaller order (where there is no longer unique factorization in prime ideals)?
I know there still is a primary decomposition of any ideal but I don't see how to find all ideals of norm $8$ above $(2)$ in $mathbb{Z}[alpha]$ for example.
algebraic-number-theory norm ideals
asked Jan 16 at 17:02
RubiskRubisk
112
$endgroup$
$begingroup$
Does that help? What do you mean?
$endgroup$
– Rubisk
Jan 16 at 19:01
$begingroup$
Why? Couldn't an ideal of norm 100 contract to an ideal of norm 25 or something? And I am talking about the non-maximal order $mathbf{Z}[sqrt[4]{24}]$ (the cuberoot is a typo I dont know how to fix).
$endgroup$
– Rubisk
Jan 19 at 21:38
$begingroup$
Assuming the class group is trivial?
$endgroup$
– Rubisk
Jan 19 at 21:57
add a comment |
$begingroup$
I have found that the ring of integers is $mathbb{Z}[alpha, alpha^3/4]$ where $alpha = sqrt[4]{24}$. I also know that in the ring of integers $(5)$ factors as two ideals of norm $25$, and $(2)$ factors as $(2, alpha)^4$ hence there are 2 ideals of norm $100$. How does this help me find the ideals of norm $100$ in the smaller order (where there is no longer unique factorization in prime ideals)?
I know there still is a primary decomposition of any ideal but I don't see how to find all ideals of norm $8$ above $(2)$ in $mathbb{Z}[alpha]$ for example.
algebraic-number-theory norm ideals
asked Jan 16 at 17:02
RubiskRubisk
112
$endgroup$
I have found that the ring of integers is $mathbb{Z}[alpha, alpha^3/4]$ where $alpha = sqrt[4]{24}$. I also know that in the ring of integers $(5)$ factors as two ideals of norm $25$, and $(2)$ factors as $(2, alpha)^4$ hence there are 2 ideals of norm $100$. How does this help me find the ideals of norm $100$ in the smaller order (where there is no longer unique factorization in prime ideals)?
I know there still is a primary decomposition of any ideal but I don't see how to find all ideals of norm $8$ above $(2)$ in $mathbb{Z}[alpha]$ for example.
algebraic-number-theory norm ideals
algebraic-number-theory norm ideals
asked Jan 16 at 17:02
RubiskRubisk
112
asked Jan 16 at 17:02
RubiskRubisk
112
edited Jan 19 at 21:39
Rubisk
asked Jan 16 at 17:02
RubiskRubisk
112
asked Jan 16 at 17:02
RubiskRubisk
112
asked Jan 16 at 17:02
RubiskRubisk
112
112
$begingroup$
Does that help? What do you mean?
$endgroup$
– Rubisk
Jan 16 at 19:01
$begingroup$
Why? Couldn't an ideal of norm 100 contract to an ideal of norm 25 or something? And I am talking about the non-maximal order $mathbf{Z}[sqrt[4]{24}]$ (the cuberoot is a typo I dont know how to fix).
$endgroup$
– Rubisk
Jan 19 at 21:38
$begingroup$
Assuming the class group is trivial?
$endgroup$
– Rubisk
Jan 19 at 21:57
add a comment |
$begingroup$
Does that help? What do you mean?
$endgroup$
– Rubisk
Jan 16 at 19:01
$begingroup$
Why? Couldn't an ideal of norm 100 contract to an ideal of norm 25 or something? And I am talking about the non-maximal order $mathbf{Z}[sqrt[4]{24}]$ (the cuberoot is a typo I dont know how to fix).
$endgroup$
– Rubisk
Jan 19 at 21:38
$begingroup$
Assuming the class group is trivial?
$endgroup$
– Rubisk
Jan 19 at 21:57
$begingroup$
Does that help? What do you mean?
$endgroup$
– Rubisk
Jan 16 at 19:01
$begingroup$
Does that help? What do you mean?
$endgroup$
– Rubisk
Jan 16 at 19:01
$begingroup$
Why? Couldn't an ideal of norm 100 contract to an ideal of norm 25 or something? And I am talking about the non-maximal order $mathbf{Z}[sqrt[4]{24}]$ (the cuberoot is a typo I dont know how to fix).
$endgroup$
– Rubisk
Jan 19 at 21:38
$begingroup$
Why? Couldn't an ideal of norm 100 contract to an ideal of norm 25 or something? And I am talking about the non-maximal order $mathbf{Z}[sqrt[4]{24}]$ (the cuberoot is a typo I dont know how to fix).
$endgroup$
– Rubisk
Jan 19 at 21:38
$begingroup$
Assuming the class group is trivial?
$endgroup$
– Rubisk
Jan 19 at 21:57
$begingroup$
Assuming the class group is trivial?
$endgroup$
– Rubisk
Jan 19 at 21:57
add a 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%2f3075985%2fconsider-the-order-mathbbz-sqrt424-find-all-ideals-of-norm-100%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.
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%2f3075985%2fconsider-the-order-mathbbz-sqrt424-find-all-ideals-of-norm-100%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
$begingroup$
Does that help? What do you mean?
$endgroup$
– Rubisk
Jan 16 at 19:01
$begingroup$
Why? Couldn't an ideal of norm 100 contract to an ideal of norm 25 or something? And I am talking about the non-maximal order $mathbf{Z}[sqrt[4]{24}]$ (the cuberoot is a typo I dont know how to fix).
$endgroup$
– Rubisk
Jan 19 at 21:38
$begingroup$
Assuming the class group is trivial?
$endgroup$
– Rubisk
Jan 19 at 21:57