Is it true that he polynomial $frac{x^p- 1}{x-1}$ ($p$ is prime) is irreducible in $mathbb{F}_2[x]$ iff $p$...
0
$begingroup$
Is it true that the polynomial $frac{x^p- 1}{x-1}$ ($p$ is prime) is irreducible in $mathbb{F}_2[x]$ iff $p$ is prime?
I know it will be true in $mathbb Q[x]$. Can anyone please help me to understand what happens in $mathbb{F}_2[x]$?
abstract-algebra polynomials ring-theory finite-fields irreducible-polynomials
edited Dec 14 '18 at 21:27
Batominovski
33.1k33293
asked Dec 14 '18 at 15:28
cmicmi
1,121312
$endgroup$
|
show 5 more comments
0
$begingroup$
Is it true that the polynomial $frac{x^p- 1}{x-1}$ ($p$ is prime) is irreducible in $mathbb{F}_2[x]$ iff $p$ is prime?
I know it will be true in $mathbb Q[x]$. Can anyone please help me to understand what happens in $mathbb{F}_2[x]$?
abstract-algebra polynomials ring-theory finite-fields irreducible-polynomials
edited Dec 14 '18 at 21:27
Batominovski
33.1k33293
asked Dec 14 '18 at 15:28
cmicmi
1,121312
$endgroup$
$begingroup$
Are you sure that "($p$ is prime)" is supposed to be there? Should it, perhaps, be "($p$ is a natural number)"?
$endgroup$
– Arthur
Dec 14 '18 at 15:30
$begingroup$
en.wikipedia.org/wiki/Cyclotomic_polynomial
$endgroup$
– vadim123
Dec 14 '18 at 15:33
$begingroup$
Is the statement true for any natural number?@Arthur
$endgroup$
– cmi
Dec 14 '18 at 15:35
2
$begingroup$
This is false when $p=7$. For prime $p> 2$, it is true if and only if $2$ is a generator mod $p$.
$endgroup$
– KCd
Dec 14 '18 at 15:43
$begingroup$
"2 is a generator mod p" - means?@KCd
$endgroup$
– cmi
Dec 14 '18 at 15:45
|
show 5 more comments
0
0
0
$begingroup$
Is it true that the polynomial $frac{x^p- 1}{x-1}$ ($p$ is prime) is irreducible in $mathbb{F}_2[x]$ iff $p$ is prime?
I know it will be true in $mathbb Q[x]$. Can anyone please help me to understand what happens in $mathbb{F}_2[x]$?
abstract-algebra polynomials ring-theory finite-fields irreducible-polynomials
edited Dec 14 '18 at 21:27
Batominovski
33.1k33293
asked Dec 14 '18 at 15:28
cmicmi
1,121312
$endgroup$
Is it true that the polynomial $frac{x^p- 1}{x-1}$ ($p$ is prime) is irreducible in $mathbb{F}_2[x]$ iff $p$ is prime?
I know it will be true in $mathbb Q[x]$. Can anyone please help me to understand what happens in $mathbb{F}_2[x]$?
abstract-algebra polynomials ring-theory finite-fields irreducible-polynomials
abstract-algebra polynomials ring-theory finite-fields irreducible-polynomials
edited Dec 14 '18 at 21:27
Batominovski
33.1k33293
asked Dec 14 '18 at 15:28
cmicmi
1,121312
edited Dec 14 '18 at 21:27
Batominovski
33.1k33293
asked Dec 14 '18 at 15:28
cmicmi
1,121312
edited Dec 14 '18 at 21:27
Batominovski
33.1k33293
edited Dec 14 '18 at 21:27
Batominovski
33.1k33293
edited Dec 14 '18 at 21:27
Batominovski
33.1k33293
33.1k33293
asked Dec 14 '18 at 15:28
cmicmi
1,121312
asked Dec 14 '18 at 15:28
cmicmi
1,121312
asked Dec 14 '18 at 15:28
cmicmi
1,121312
1,121312
$begingroup$
Are you sure that "($p$ is prime)" is supposed to be there? Should it, perhaps, be "($p$ is a natural number)"?
$endgroup$
– Arthur
Dec 14 '18 at 15:30
$begingroup$
en.wikipedia.org/wiki/Cyclotomic_polynomial
$endgroup$
– vadim123
Dec 14 '18 at 15:33
$begingroup$
Is the statement true for any natural number?@Arthur
$endgroup$
– cmi
Dec 14 '18 at 15:35
2
$begingroup$
This is false when $p=7$. For prime $p> 2$, it is true if and only if $2$ is a generator mod $p$.
$endgroup$
– KCd
Dec 14 '18 at 15:43
$begingroup$
"2 is a generator mod p" - means?@KCd
$endgroup$
– cmi
Dec 14 '18 at 15:45
|
show 5 more comments
$begingroup$
Are you sure that "($p$ is prime)" is supposed to be there? Should it, perhaps, be "($p$ is a natural number)"?
$endgroup$
– Arthur
Dec 14 '18 at 15:30
$begingroup$
en.wikipedia.org/wiki/Cyclotomic_polynomial
$endgroup$
– vadim123
Dec 14 '18 at 15:33
$begingroup$
Is the statement true for any natural number?@Arthur
$endgroup$
– cmi
Dec 14 '18 at 15:35
2
$begingroup$
This is false when $p=7$. For prime $p> 2$, it is true if and only if $2$ is a generator mod $p$.
$endgroup$
– KCd
Dec 14 '18 at 15:43
$begingroup$
"2 is a generator mod p" - means?@KCd
$endgroup$
– cmi
Dec 14 '18 at 15:45
$begingroup$
Are you sure that "($p$ is prime)" is supposed to be there? Should it, perhaps, be "($p$ is a natural number)"?
$endgroup$
– Arthur
Dec 14 '18 at 15:30
$begingroup$
Are you sure that "($p$ is prime)" is supposed to be there? Should it, perhaps, be "($p$ is a natural number)"?
$endgroup$
– Arthur
Dec 14 '18 at 15:30
$begingroup$
en.wikipedia.org/wiki/Cyclotomic_polynomial
$endgroup$
– vadim123
Dec 14 '18 at 15:33
$begingroup$
en.wikipedia.org/wiki/Cyclotomic_polynomial
$endgroup$
– vadim123
Dec 14 '18 at 15:33
$begingroup$
Is the statement true for any natural number?@Arthur
$endgroup$
– cmi
Dec 14 '18 at 15:35
$begingroup$
Is the statement true for any natural number?@Arthur
$endgroup$
– cmi
Dec 14 '18 at 15:35
2
2
$begingroup$
This is false when $p=7$. For prime $p> 2$, it is true if and only if $2$ is a generator mod $p$.
$endgroup$
– KCd
Dec 14 '18 at 15:43
$begingroup$
This is false when $p=7$. For prime $p> 2$, it is true if and only if $2$ is a generator mod $p$.
$endgroup$
– KCd
Dec 14 '18 at 15:43
$begingroup$
"2 is a generator mod p" - means?@KCd
$endgroup$
– cmi
Dec 14 '18 at 15:45
$begingroup$
"2 is a generator mod p" - means?@KCd
$endgroup$
– cmi
Dec 14 '18 at 15:45
|
show 5 more comments
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
});
}
});
draft saved
draft discarded
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%2f3039498%2fis-it-true-that-he-polynomial-fracxp-1x-1-p-is-prime-is-irreducible%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
draft saved
draft discarded
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.
draft saved
draft discarded
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%2f3039498%2fis-it-true-that-he-polynomial-fracxp-1x-1-p-is-prime-is-irreducible%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
$begingroup$
Are you sure that "($p$ is prime)" is supposed to be there? Should it, perhaps, be "($p$ is a natural number)"?
$endgroup$
– Arthur
Dec 14 '18 at 15:30
$begingroup$
en.wikipedia.org/wiki/Cyclotomic_polynomial
$endgroup$
– vadim123
Dec 14 '18 at 15:33
$begingroup$
Is the statement true for any natural number?@Arthur
$endgroup$
– cmi
Dec 14 '18 at 15:35
2
$begingroup$
This is false when $p=7$. For prime $p> 2$, it is true if and only if $2$ is a generator mod $p$.
$endgroup$
– KCd
Dec 14 '18 at 15:43
$begingroup$
"2 is a generator mod p" - means?@KCd
$endgroup$
– cmi
Dec 14 '18 at 15:45