factors of $X^{p^n}-X$












1














I'm my study of Galois theory I have been struggling with the following proposition without much success:




The polynomial $X^{p^n}-X$ is precisely the product of all the distinct irreducible polynomials in $mathbb{F}_p[X]$ of degree $d$ where $d$ runs through all divisors of $n$




For example, $X^{p^6}-X$ is factorized by a polynomial of degree 1, 2, 3 and 6? It is posible to obtain them explicitly?










share|cite|improve this question
























  • You probably mean mod $p$. And the answer depends in $p$. Start here: Cyclotomic polynomial - Wikipedia
    – lhf
    Dec 6 at 16:59












  • @lhf yep edited for $mathbb{F}_p[X]$. I'm going to read about your link. Thanks.
    – UnPerrito
    Dec 6 at 17:02










  • If $mathbb{F}_p(alpha) = mathbb{F}_{p^6}$ then what is the minimal polynomial of $alpha$ over $mathbb{F}_p$ ?
    – reuns
    Dec 6 at 17:12










  • @reuns as far I know, it must be a polynomial of degree 6
    – UnPerrito
    Dec 6 at 17:18






  • 1




    Finding those factors is possible but somewhat taxing. If the only input you are given is: Factor $x^{p^n}-x$! then I would go with the Cantor-Zassenhaus algorithm. Works for any polynomial over any finite field! But, if $p$ and/or $n$ are largish it will be a lot of work. This toy example I did when learning about C-Z will show what to expect.
    – Jyrki Lahtonen
    Dec 6 at 18:58
















1














I'm my study of Galois theory I have been struggling with the following proposition without much success:




The polynomial $X^{p^n}-X$ is precisely the product of all the distinct irreducible polynomials in $mathbb{F}_p[X]$ of degree $d$ where $d$ runs through all divisors of $n$




For example, $X^{p^6}-X$ is factorized by a polynomial of degree 1, 2, 3 and 6? It is posible to obtain them explicitly?










share|cite|improve this question
























  • You probably mean mod $p$. And the answer depends in $p$. Start here: Cyclotomic polynomial - Wikipedia
    – lhf
    Dec 6 at 16:59












  • @lhf yep edited for $mathbb{F}_p[X]$. I'm going to read about your link. Thanks.
    – UnPerrito
    Dec 6 at 17:02










  • If $mathbb{F}_p(alpha) = mathbb{F}_{p^6}$ then what is the minimal polynomial of $alpha$ over $mathbb{F}_p$ ?
    – reuns
    Dec 6 at 17:12










  • @reuns as far I know, it must be a polynomial of degree 6
    – UnPerrito
    Dec 6 at 17:18






  • 1




    Finding those factors is possible but somewhat taxing. If the only input you are given is: Factor $x^{p^n}-x$! then I would go with the Cantor-Zassenhaus algorithm. Works for any polynomial over any finite field! But, if $p$ and/or $n$ are largish it will be a lot of work. This toy example I did when learning about C-Z will show what to expect.
    – Jyrki Lahtonen
    Dec 6 at 18:58














1












1








1







I'm my study of Galois theory I have been struggling with the following proposition without much success:




The polynomial $X^{p^n}-X$ is precisely the product of all the distinct irreducible polynomials in $mathbb{F}_p[X]$ of degree $d$ where $d$ runs through all divisors of $n$




For example, $X^{p^6}-X$ is factorized by a polynomial of degree 1, 2, 3 and 6? It is posible to obtain them explicitly?










share|cite|improve this question















I'm my study of Galois theory I have been struggling with the following proposition without much success:




The polynomial $X^{p^n}-X$ is precisely the product of all the distinct irreducible polynomials in $mathbb{F}_p[X]$ of degree $d$ where $d$ runs through all divisors of $n$




For example, $X^{p^6}-X$ is factorized by a polynomial of degree 1, 2, 3 and 6? It is posible to obtain them explicitly?







galois-theory finite-fields irreducible-polynomials






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 6 at 17:00

























asked Dec 6 at 16:55









UnPerrito

908




908












  • You probably mean mod $p$. And the answer depends in $p$. Start here: Cyclotomic polynomial - Wikipedia
    – lhf
    Dec 6 at 16:59












  • @lhf yep edited for $mathbb{F}_p[X]$. I'm going to read about your link. Thanks.
    – UnPerrito
    Dec 6 at 17:02










  • If $mathbb{F}_p(alpha) = mathbb{F}_{p^6}$ then what is the minimal polynomial of $alpha$ over $mathbb{F}_p$ ?
    – reuns
    Dec 6 at 17:12










  • @reuns as far I know, it must be a polynomial of degree 6
    – UnPerrito
    Dec 6 at 17:18






  • 1




    Finding those factors is possible but somewhat taxing. If the only input you are given is: Factor $x^{p^n}-x$! then I would go with the Cantor-Zassenhaus algorithm. Works for any polynomial over any finite field! But, if $p$ and/or $n$ are largish it will be a lot of work. This toy example I did when learning about C-Z will show what to expect.
    – Jyrki Lahtonen
    Dec 6 at 18:58


















  • You probably mean mod $p$. And the answer depends in $p$. Start here: Cyclotomic polynomial - Wikipedia
    – lhf
    Dec 6 at 16:59












  • @lhf yep edited for $mathbb{F}_p[X]$. I'm going to read about your link. Thanks.
    – UnPerrito
    Dec 6 at 17:02










  • If $mathbb{F}_p(alpha) = mathbb{F}_{p^6}$ then what is the minimal polynomial of $alpha$ over $mathbb{F}_p$ ?
    – reuns
    Dec 6 at 17:12










  • @reuns as far I know, it must be a polynomial of degree 6
    – UnPerrito
    Dec 6 at 17:18






  • 1




    Finding those factors is possible but somewhat taxing. If the only input you are given is: Factor $x^{p^n}-x$! then I would go with the Cantor-Zassenhaus algorithm. Works for any polynomial over any finite field! But, if $p$ and/or $n$ are largish it will be a lot of work. This toy example I did when learning about C-Z will show what to expect.
    – Jyrki Lahtonen
    Dec 6 at 18:58
















You probably mean mod $p$. And the answer depends in $p$. Start here: Cyclotomic polynomial - Wikipedia
– lhf
Dec 6 at 16:59






You probably mean mod $p$. And the answer depends in $p$. Start here: Cyclotomic polynomial - Wikipedia
– lhf
Dec 6 at 16:59














@lhf yep edited for $mathbb{F}_p[X]$. I'm going to read about your link. Thanks.
– UnPerrito
Dec 6 at 17:02




@lhf yep edited for $mathbb{F}_p[X]$. I'm going to read about your link. Thanks.
– UnPerrito
Dec 6 at 17:02












If $mathbb{F}_p(alpha) = mathbb{F}_{p^6}$ then what is the minimal polynomial of $alpha$ over $mathbb{F}_p$ ?
– reuns
Dec 6 at 17:12




If $mathbb{F}_p(alpha) = mathbb{F}_{p^6}$ then what is the minimal polynomial of $alpha$ over $mathbb{F}_p$ ?
– reuns
Dec 6 at 17:12












@reuns as far I know, it must be a polynomial of degree 6
– UnPerrito
Dec 6 at 17:18




@reuns as far I know, it must be a polynomial of degree 6
– UnPerrito
Dec 6 at 17:18




1




1




Finding those factors is possible but somewhat taxing. If the only input you are given is: Factor $x^{p^n}-x$! then I would go with the Cantor-Zassenhaus algorithm. Works for any polynomial over any finite field! But, if $p$ and/or $n$ are largish it will be a lot of work. This toy example I did when learning about C-Z will show what to expect.
– Jyrki Lahtonen
Dec 6 at 18:58




Finding those factors is possible but somewhat taxing. If the only input you are given is: Factor $x^{p^n}-x$! then I would go with the Cantor-Zassenhaus algorithm. Works for any polynomial over any finite field! But, if $p$ and/or $n$ are largish it will be a lot of work. This toy example I did when learning about C-Z will show what to expect.
– Jyrki Lahtonen
Dec 6 at 18:58















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


















StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3028761%2ffactors-of-xpn-x%23new-answer', 'question_page');
}
);

Post as a guest















Required, but never shown






























active

oldest

votes













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.





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.




draft saved


draft discarded














StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3028761%2ffactors-of-xpn-x%23new-answer', 'question_page');
}
);

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







Popular posts from this blog

Bressuire

Cabo Verde

Gyllenstierna