Adjoining an element with given minimal polynomial to a DVR of characteristic p.
up vote
1
down vote
favorite
Let $A$ be a DVR of characteristic $p$, with $pi$ a uniformising parameter, with $K=frac(A)$ the field of fractions. Consider the extension $L=K(alpha)$ where $alpha$ has minimal polynomial $y^p+pi^b y+pi^c$, with $0<bleq c$. The problem is to show the existence of an element of $L$ with minimal polynomial $y^p+upi y+vpi^{p(c-b)}$, with $u,vin A^*$.
I think this is going to be manipulating the known minimal polynomial of $alpha$ to produce such another element, eg $alpha pi^k+pi^l$, for $k,linmathbb{Z}$, possibly using the Frobenius map.
I wasn't able to make this work however, so it may require a more sophisticated idea. Any hints would be much appreciated.
abstract-algebra algebraic-number-theory valuation-theory local-rings
asked Nov 29 at 5:49
user277182
381210
add a comment |
up vote
1
down vote
favorite
Let $A$ be a DVR of characteristic $p$, with $pi$ a uniformising parameter, with $K=frac(A)$ the field of fractions. Consider the extension $L=K(alpha)$ where $alpha$ has minimal polynomial $y^p+pi^b y+pi^c$, with $0<bleq c$. The problem is to show the existence of an element of $L$ with minimal polynomial $y^p+upi y+vpi^{p(c-b)}$, with $u,vin A^*$.
I think this is going to be manipulating the known minimal polynomial of $alpha$ to produce such another element, eg $alpha pi^k+pi^l$, for $k,linmathbb{Z}$, possibly using the Frobenius map.
I wasn't able to make this work however, so it may require a more sophisticated idea. Any hints would be much appreciated.
abstract-algebra algebraic-number-theory valuation-theory local-rings
asked Nov 29 at 5:49
user277182
381210
I think your attempt is good. For the simplest case $b=c=1$, setting $alpha+1$ as the new element works. For the general case, I would try $k=l :=c-b$, in other words, the new element would be $pi^{c-b} cdot(alpha+1)$.
– Torsten Schoeneberg
Dec 4 at 2:01
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $A$ be a DVR of characteristic $p$, with $pi$ a uniformising parameter, with $K=frac(A)$ the field of fractions. Consider the extension $L=K(alpha)$ where $alpha$ has minimal polynomial $y^p+pi^b y+pi^c$, with $0<bleq c$. The problem is to show the existence of an element of $L$ with minimal polynomial $y^p+upi y+vpi^{p(c-b)}$, with $u,vin A^*$.
I think this is going to be manipulating the known minimal polynomial of $alpha$ to produce such another element, eg $alpha pi^k+pi^l$, for $k,linmathbb{Z}$, possibly using the Frobenius map.
I wasn't able to make this work however, so it may require a more sophisticated idea. Any hints would be much appreciated.
abstract-algebra algebraic-number-theory valuation-theory local-rings
asked Nov 29 at 5:49
user277182
381210
Let $A$ be a DVR of characteristic $p$, with $pi$ a uniformising parameter, with $K=frac(A)$ the field of fractions. Consider the extension $L=K(alpha)$ where $alpha$ has minimal polynomial $y^p+pi^b y+pi^c$, with $0<bleq c$. The problem is to show the existence of an element of $L$ with minimal polynomial $y^p+upi y+vpi^{p(c-b)}$, with $u,vin A^*$.
I think this is going to be manipulating the known minimal polynomial of $alpha$ to produce such another element, eg $alpha pi^k+pi^l$, for $k,linmathbb{Z}$, possibly using the Frobenius map.
I wasn't able to make this work however, so it may require a more sophisticated idea. Any hints would be much appreciated.
abstract-algebra algebraic-number-theory valuation-theory local-rings
abstract-algebra algebraic-number-theory valuation-theory local-rings
asked Nov 29 at 5:49
user277182
381210
asked Nov 29 at 5:49
user277182
381210
asked Nov 29 at 5:49
user277182
381210
asked Nov 29 at 5:49
user277182
381210
asked Nov 29 at 5:49
user277182
381210
381210
I think your attempt is good. For the simplest case $b=c=1$, setting $alpha+1$ as the new element works. For the general case, I would try $k=l :=c-b$, in other words, the new element would be $pi^{c-b} cdot(alpha+1)$.
– Torsten Schoeneberg
Dec 4 at 2:01
add a comment |
I think your attempt is good. For the simplest case $b=c=1$, setting $alpha+1$ as the new element works. For the general case, I would try $k=l :=c-b$, in other words, the new element would be $pi^{c-b} cdot(alpha+1)$.
– Torsten Schoeneberg
Dec 4 at 2:01
I think your attempt is good. For the simplest case $b=c=1$, setting $alpha+1$ as the new element works. For the general case, I would try $k=l :=c-b$, in other words, the new element would be $pi^{c-b} cdot(alpha+1)$.
– Torsten Schoeneberg
Dec 4 at 2:01
I think your attempt is good. For the simplest case $b=c=1$, setting $alpha+1$ as the new element works. For the general case, I would try $k=l :=c-b$, in other words, the new element would be $pi^{c-b} cdot(alpha+1)$.
– Torsten Schoeneberg
Dec 4 at 2:01
add a comment |
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%2f3018242%2fadjoining-an-element-with-given-minimal-polynomial-to-a-dvr-of-characteristic-p%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
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%2f3018242%2fadjoining-an-element-with-given-minimal-polynomial-to-a-dvr-of-characteristic-p%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
I think your attempt is good. For the simplest case $b=c=1$, setting $alpha+1$ as the new element works. For the general case, I would try $k=l :=c-b$, in other words, the new element would be $pi^{c-b} cdot(alpha+1)$.
– Torsten Schoeneberg
Dec 4 at 2:01