factors of $X^{p^n}-X$
1
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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
asked Dec 6 '18 at 16:55
UnPerritoUnPerrito
908
$endgroup$
|
show 2 more comments
1
$begingroup$
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
asked Dec 6 '18 at 16:55
UnPerritoUnPerrito
908
$endgroup$
$begingroup$
You probably mean mod $p$. And the answer depends in $p$. Start here: Cyclotomic polynomial - Wikipedia
$endgroup$
– lhf
Dec 6 '18 at 16:59
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@lhf yep edited for $mathbb{F}_p[X]$. I'm going to read about your link. Thanks.
$endgroup$
– UnPerrito
Dec 6 '18 at 17:02
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If $mathbb{F}_p(alpha) = mathbb{F}_{p^6}$ then what is the minimal polynomial of $alpha$ over $mathbb{F}_p$ ?
$endgroup$
– reuns
Dec 6 '18 at 17:12
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@reuns as far I know, it must be a polynomial of degree 6
$endgroup$
– UnPerrito
Dec 6 '18 at 17:18
1
$begingroup$
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.
$endgroup$
– Jyrki Lahtonen
Dec 6 '18 at 18:58
|
show 2 more comments
1
1
1
$begingroup$
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
asked Dec 6 '18 at 16:55
UnPerritoUnPerrito
908
$endgroup$
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
galois-theory finite-fields irreducible-polynomials
asked Dec 6 '18 at 16:55
UnPerritoUnPerrito
908
asked Dec 6 '18 at 16:55
UnPerritoUnPerrito
908
edited Dec 6 '18 at 17:00
UnPerrito
asked Dec 6 '18 at 16:55
UnPerritoUnPerrito
908
asked Dec 6 '18 at 16:55
UnPerritoUnPerrito
908
asked Dec 6 '18 at 16:55
UnPerritoUnPerrito
908
908
$begingroup$
You probably mean mod $p$. And the answer depends in $p$. Start here: Cyclotomic polynomial - Wikipedia
$endgroup$
– lhf
Dec 6 '18 at 16:59
$begingroup$
@lhf yep edited for $mathbb{F}_p[X]$. I'm going to read about your link. Thanks.
$endgroup$
– UnPerrito
Dec 6 '18 at 17:02
$begingroup$
If $mathbb{F}_p(alpha) = mathbb{F}_{p^6}$ then what is the minimal polynomial of $alpha$ over $mathbb{F}_p$ ?
$endgroup$
– reuns
Dec 6 '18 at 17:12
$begingroup$
@reuns as far I know, it must be a polynomial of degree 6
$endgroup$
– UnPerrito
Dec 6 '18 at 17:18
1
$begingroup$
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.
$endgroup$
– Jyrki Lahtonen
Dec 6 '18 at 18:58
|
show 2 more comments
$begingroup$
You probably mean mod $p$. And the answer depends in $p$. Start here: Cyclotomic polynomial - Wikipedia
$endgroup$
– lhf
Dec 6 '18 at 16:59
$begingroup$
@lhf yep edited for $mathbb{F}_p[X]$. I'm going to read about your link. Thanks.
$endgroup$
– UnPerrito
Dec 6 '18 at 17:02
$begingroup$
If $mathbb{F}_p(alpha) = mathbb{F}_{p^6}$ then what is the minimal polynomial of $alpha$ over $mathbb{F}_p$ ?
$endgroup$
– reuns
Dec 6 '18 at 17:12
$begingroup$
@reuns as far I know, it must be a polynomial of degree 6
$endgroup$
– UnPerrito
Dec 6 '18 at 17:18
1
$begingroup$
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.
$endgroup$
– Jyrki Lahtonen
Dec 6 '18 at 18:58
$begingroup$
You probably mean mod $p$. And the answer depends in $p$. Start here: Cyclotomic polynomial - Wikipedia
$endgroup$
– lhf
Dec 6 '18 at 16:59
$begingroup$
You probably mean mod $p$. And the answer depends in $p$. Start here: Cyclotomic polynomial - Wikipedia
$endgroup$
– lhf
Dec 6 '18 at 16:59
$begingroup$
@lhf yep edited for $mathbb{F}_p[X]$. I'm going to read about your link. Thanks.
$endgroup$
– UnPerrito
Dec 6 '18 at 17:02
$begingroup$
@lhf yep edited for $mathbb{F}_p[X]$. I'm going to read about your link. Thanks.
$endgroup$
– UnPerrito
Dec 6 '18 at 17:02
$begingroup$
If $mathbb{F}_p(alpha) = mathbb{F}_{p^6}$ then what is the minimal polynomial of $alpha$ over $mathbb{F}_p$ ?
$endgroup$
– reuns
Dec 6 '18 at 17:12
$begingroup$
If $mathbb{F}_p(alpha) = mathbb{F}_{p^6}$ then what is the minimal polynomial of $alpha$ over $mathbb{F}_p$ ?
$endgroup$
– reuns
Dec 6 '18 at 17:12
$begingroup$
@reuns as far I know, it must be a polynomial of degree 6
$endgroup$
– UnPerrito
Dec 6 '18 at 17:18
$begingroup$
@reuns as far I know, it must be a polynomial of degree 6
$endgroup$
– UnPerrito
Dec 6 '18 at 17:18
1
1
$begingroup$
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.
$endgroup$
– Jyrki Lahtonen
Dec 6 '18 at 18:58
$begingroup$
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.
$endgroup$
– Jyrki Lahtonen
Dec 6 '18 at 18:58
|
show 2 more comments
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$begingroup$
You probably mean mod $p$. And the answer depends in $p$. Start here: Cyclotomic polynomial - Wikipedia
$endgroup$
– lhf
Dec 6 '18 at 16:59
$begingroup$
@lhf yep edited for $mathbb{F}_p[X]$. I'm going to read about your link. Thanks.
$endgroup$
– UnPerrito
Dec 6 '18 at 17:02
$begingroup$
If $mathbb{F}_p(alpha) = mathbb{F}_{p^6}$ then what is the minimal polynomial of $alpha$ over $mathbb{F}_p$ ?
$endgroup$
– reuns
Dec 6 '18 at 17:12
$begingroup$
@reuns as far I know, it must be a polynomial of degree 6
$endgroup$
– UnPerrito
Dec 6 '18 at 17:18
1
$begingroup$
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.
$endgroup$
– Jyrki Lahtonen
Dec 6 '18 at 18:58