Mixing
Some questions regarding the proof of Luroth theorem given by G. Bergman
G. Bergman gave a series of exercises leading to the proof of Luroth's Theorem: https://math.berkeley.edu/~gbergman/grad.hndts/Luroth.ps
My concern lays in problem (7) and (8).
$L$ is any intermediate field: $k subsetneq L subset k(t)$, where $t$ is transcendental over $k$. $u=P(t)/Q(t)$ is an element of $L-k$.
Please refer to the pdf file for other details.
Show that if $P(x)-uQ(x)$ is divisible in $L[x]$ by an element of $k[x]$, then this element must divide both $P(x)$ and $Q(x)$.
how to prove it bothers me; my thought is like this: $ {1,u} subset B$, where $B$ is a basis of $L$ over $k$. Then $L[x] = k[u,..][x]$. After we embed $k[x]$ into $L[x]$, $u$ clearly is not divisible by any element from $k[x]$ then $P(x)$ and $Q(x)$ has to be divided by the element respectively. Am I correct?
In addition, is there any theorem leading to such result that if $t$ has an identical minimal polynomial over $k(u)$ and $L$ while $k(u) subset L$, then $L=k(u)$?
abstract-algebra field-theory divisibility extension-field
asked Nov 20 '18 at 0:16
Alex Gong
11
add a comment |
G. Bergman gave a series of exercises leading to the proof of Luroth's Theorem: https://math.berkeley.edu/~gbergman/grad.hndts/Luroth.ps
My concern lays in problem (7) and (8).
$L$ is any intermediate field: $k subsetneq L subset k(t)$, where $t$ is transcendental over $k$. $u=P(t)/Q(t)$ is an element of $L-k$.
Please refer to the pdf file for other details.
Show that if $P(x)-uQ(x)$ is divisible in $L[x]$ by an element of $k[x]$, then this element must divide both $P(x)$ and $Q(x)$.
how to prove it bothers me; my thought is like this: $ {1,u} subset B$, where $B$ is a basis of $L$ over $k$. Then $L[x] = k[u,..][x]$. After we embed $k[x]$ into $L[x]$, $u$ clearly is not divisible by any element from $k[x]$ then $P(x)$ and $Q(x)$ has to be divided by the element respectively. Am I correct?
In addition, is there any theorem leading to such result that if $t$ has an identical minimal polynomial over $k(u)$ and $L$ while $k(u) subset L$, then $L=k(u)$?
abstract-algebra field-theory divisibility extension-field
asked Nov 20 '18 at 0:16
Alex Gong
11
What are $P$ and $Q$? Polynomials in $kleft[xright]$ or in $Lleft[xright]$ ?
– darij grinberg
Nov 20 '18 at 4:23
Also your last question sounds weird to me. Could it be the $subset$ sign should be $supset$?
– darij grinberg
Nov 20 '18 at 4:23
i suggest that if possible, you shall read the pdf file. The sign is correct, since $u$ is in $L-k$.
– Alex Gong
Nov 20 '18 at 4:48
add a comment |
G. Bergman gave a series of exercises leading to the proof of Luroth's Theorem: https://math.berkeley.edu/~gbergman/grad.hndts/Luroth.ps
My concern lays in problem (7) and (8).
$L$ is any intermediate field: $k subsetneq L subset k(t)$, where $t$ is transcendental over $k$. $u=P(t)/Q(t)$ is an element of $L-k$.
Please refer to the pdf file for other details.
Show that if $P(x)-uQ(x)$ is divisible in $L[x]$ by an element of $k[x]$, then this element must divide both $P(x)$ and $Q(x)$.
how to prove it bothers me; my thought is like this: $ {1,u} subset B$, where $B$ is a basis of $L$ over $k$. Then $L[x] = k[u,..][x]$. After we embed $k[x]$ into $L[x]$, $u$ clearly is not divisible by any element from $k[x]$ then $P(x)$ and $Q(x)$ has to be divided by the element respectively. Am I correct?
In addition, is there any theorem leading to such result that if $t$ has an identical minimal polynomial over $k(u)$ and $L$ while $k(u) subset L$, then $L=k(u)$?
abstract-algebra field-theory divisibility extension-field
asked Nov 20 '18 at 0:16
Alex Gong
11
G. Bergman gave a series of exercises leading to the proof of Luroth's Theorem: https://math.berkeley.edu/~gbergman/grad.hndts/Luroth.ps
My concern lays in problem (7) and (8).
$L$ is any intermediate field: $k subsetneq L subset k(t)$, where $t$ is transcendental over $k$. $u=P(t)/Q(t)$ is an element of $L-k$.
Please refer to the pdf file for other details.
Show that if $P(x)-uQ(x)$ is divisible in $L[x]$ by an element of $k[x]$, then this element must divide both $P(x)$ and $Q(x)$.
how to prove it bothers me; my thought is like this: $ {1,u} subset B$, where $B$ is a basis of $L$ over $k$. Then $L[x] = k[u,..][x]$. After we embed $k[x]$ into $L[x]$, $u$ clearly is not divisible by any element from $k[x]$ then $P(x)$ and $Q(x)$ has to be divided by the element respectively. Am I correct?
In addition, is there any theorem leading to such result that if $t$ has an identical minimal polynomial over $k(u)$ and $L$ while $k(u) subset L$, then $L=k(u)$?
abstract-algebra field-theory divisibility extension-field
abstract-algebra field-theory divisibility extension-field
asked Nov 20 '18 at 0:16
Alex Gong
11
asked Nov 20 '18 at 0:16
Alex Gong
11
edited Nov 20 '18 at 6:17
asked Nov 20 '18 at 0:16
Alex Gong
11
asked Nov 20 '18 at 0:16
Alex Gong
11
asked Nov 20 '18 at 0:16
Alex Gong
11
11
What are $P$ and $Q$? Polynomials in $kleft[xright]$ or in $Lleft[xright]$ ?
– darij grinberg
Nov 20 '18 at 4:23
Also your last question sounds weird to me. Could it be the $subset$ sign should be $supset$?
– darij grinberg
Nov 20 '18 at 4:23
i suggest that if possible, you shall read the pdf file. The sign is correct, since $u$ is in $L-k$.
– Alex Gong
Nov 20 '18 at 4:48
add a comment |
What are $P$ and $Q$? Polynomials in $kleft[xright]$ or in $Lleft[xright]$ ?
– darij grinberg
Nov 20 '18 at 4:23
Also your last question sounds weird to me. Could it be the $subset$ sign should be $supset$?
– darij grinberg
Nov 20 '18 at 4:23
i suggest that if possible, you shall read the pdf file. The sign is correct, since $u$ is in $L-k$.
– Alex Gong
Nov 20 '18 at 4:48
What are $P$ and $Q$? Polynomials in $kleft[xright]$ or in $Lleft[xright]$ ?
– darij grinberg
Nov 20 '18 at 4:23
What are $P$ and $Q$? Polynomials in $kleft[xright]$ or in $Lleft[xright]$ ?
– darij grinberg
Nov 20 '18 at 4:23
Also your last question sounds weird to me. Could it be the $subset$ sign should be $supset$?
– darij grinberg
Nov 20 '18 at 4:23
Also your last question sounds weird to me. Could it be the $subset$ sign should be $supset$?
– darij grinberg
Nov 20 '18 at 4:23
i suggest that if possible, you shall read the pdf file. The sign is correct, since $u$ is in $L-k$.
– Alex Gong
Nov 20 '18 at 4:48
i suggest that if possible, you shall read the pdf file. The sign is correct, since $u$ is in $L-k$.
– Alex Gong
Nov 20 '18 at 4:48
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%2f3005735%2fsome-questions-regarding-the-proof-of-luroth-theorem-given-by-g-bergman%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%2f3005735%2fsome-questions-regarding-the-proof-of-luroth-theorem-given-by-g-bergman%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
What are $P$ and $Q$? Polynomials in $kleft[xright]$ or in $Lleft[xright]$ ?
– darij grinberg
Nov 20 '18 at 4:23
Also your last question sounds weird to me. Could it be the $subset$ sign should be $supset$?
– darij grinberg
Nov 20 '18 at 4:23
i suggest that if possible, you shall read the pdf file. The sign is correct, since $u$ is in $L-k$.
– Alex Gong
Nov 20 '18 at 4:48