Mixing
How to show that it is irreducible?
0
$begingroup$
Let $gcd(p,n)=1$.
Consider $;x^n-1$ over $Bbb F_p[x];$. If its splitting field is $K$ find $;[K:Bbb F_p]$.
Now $K=Bbb F_p(e^{2pi i/n})$
Also the polynomial of which $e^{2pi i/n}$ is a root is $1+x+x^2+ldots +x^{n-1}$
But How to show that it is irreducible?
Do we need to use the fact that $gcd(p,n)=1$.
Can I get some help please?
field-theory irreducible-polynomials
edited Jan 26 at 12:58
reuns
21.6k21352
$endgroup$
|
show 6 more comments
0
$begingroup$
Let $gcd(p,n)=1$.
Consider $;x^n-1$ over $Bbb F_p[x];$. If its splitting field is $K$ find $;[K:Bbb F_p]$.
Now $K=Bbb F_p(e^{2pi i/n})$
Also the polynomial of which $e^{2pi i/n}$ is a root is $1+x+x^2+ldots +x^{n-1}$
But How to show that it is irreducible?
Do we need to use the fact that $gcd(p,n)=1$.
Can I get some help please?
field-theory irreducible-polynomials
edited Jan 26 at 12:58
reuns
21.6k21352
$endgroup$
$begingroup$
@DietrichBurde;$Bbb Z_p$ means the ring $Bbb Z_n$ where $n=p$ and it is a field
$endgroup$
– user596656
Jan 26 at 12:51
$begingroup$
This finite field is usually denoted by $Bbb F_p$.
$endgroup$
– Dietrich Burde
Jan 26 at 12:53
$begingroup$
@DietrichBurde;i meant that only
$endgroup$
– user596656
Jan 26 at 12:53
$begingroup$
Possible duplicate of Irreducible cyclotomic polynomial over a field $mathbb{F}_q$?
$endgroup$
– Dietrich Burde
Jan 26 at 12:54
$begingroup$
You can use $mathbb{Z}_p$ for the cyclic group but in Galois field theory $mathbb{Z}_p$ always refers to the $p$-adic integers (it is not a field but your question still made sense there)
$endgroup$
– reuns
Jan 26 at 12:58
|
show 6 more comments
0
0
0
$begingroup$
Let $gcd(p,n)=1$.
Consider $;x^n-1$ over $Bbb F_p[x];$. If its splitting field is $K$ find $;[K:Bbb F_p]$.
Now $K=Bbb F_p(e^{2pi i/n})$
Also the polynomial of which $e^{2pi i/n}$ is a root is $1+x+x^2+ldots +x^{n-1}$
But How to show that it is irreducible?
Do we need to use the fact that $gcd(p,n)=1$.
Can I get some help please?
field-theory irreducible-polynomials
edited Jan 26 at 12:58
reuns
21.6k21352
$endgroup$
Let $gcd(p,n)=1$.
Consider $;x^n-1$ over $Bbb F_p[x];$. If its splitting field is $K$ find $;[K:Bbb F_p]$.
Now $K=Bbb F_p(e^{2pi i/n})$
Also the polynomial of which $e^{2pi i/n}$ is a root is $1+x+x^2+ldots +x^{n-1}$
But How to show that it is irreducible?
Do we need to use the fact that $gcd(p,n)=1$.
Can I get some help please?
field-theory irreducible-polynomials
field-theory irreducible-polynomials
edited Jan 26 at 12:58
reuns
21.6k21352
edited Jan 26 at 12:58
reuns
21.6k21352
edited Jan 26 at 12:58
reuns
21.6k21352
edited Jan 26 at 12:58
reuns
21.6k21352
edited Jan 26 at 12:58
reuns
21.6k21352
21.6k21352
asked Jan 26 at 12:40
user596656
$begingroup$
@DietrichBurde;$Bbb Z_p$ means the ring $Bbb Z_n$ where $n=p$ and it is a field
$endgroup$
– user596656
Jan 26 at 12:51
$begingroup$
This finite field is usually denoted by $Bbb F_p$.
$endgroup$
– Dietrich Burde
Jan 26 at 12:53
$begingroup$
@DietrichBurde;i meant that only
$endgroup$
– user596656
Jan 26 at 12:53
$begingroup$
Possible duplicate of Irreducible cyclotomic polynomial over a field $mathbb{F}_q$?
$endgroup$
– Dietrich Burde
Jan 26 at 12:54
$begingroup$
You can use $mathbb{Z}_p$ for the cyclic group but in Galois field theory $mathbb{Z}_p$ always refers to the $p$-adic integers (it is not a field but your question still made sense there)
$endgroup$
– reuns
Jan 26 at 12:58
|
show 6 more comments
$begingroup$
@DietrichBurde;$Bbb Z_p$ means the ring $Bbb Z_n$ where $n=p$ and it is a field
$endgroup$
– user596656
Jan 26 at 12:51
$begingroup$
This finite field is usually denoted by $Bbb F_p$.
$endgroup$
– Dietrich Burde
Jan 26 at 12:53
$begingroup$
@DietrichBurde;i meant that only
$endgroup$
– user596656
Jan 26 at 12:53
$begingroup$
Possible duplicate of Irreducible cyclotomic polynomial over a field $mathbb{F}_q$?
$endgroup$
– Dietrich Burde
Jan 26 at 12:54
$begingroup$
You can use $mathbb{Z}_p$ for the cyclic group but in Galois field theory $mathbb{Z}_p$ always refers to the $p$-adic integers (it is not a field but your question still made sense there)
$endgroup$
– reuns
Jan 26 at 12:58
$begingroup$
@DietrichBurde;$Bbb Z_p$ means the ring $Bbb Z_n$ where $n=p$ and it is a field
$endgroup$
– user596656
Jan 26 at 12:51
$begingroup$
@DietrichBurde;$Bbb Z_p$ means the ring $Bbb Z_n$ where $n=p$ and it is a field
$endgroup$
– user596656
Jan 26 at 12:51
$begingroup$
This finite field is usually denoted by $Bbb F_p$.
$endgroup$
– Dietrich Burde
Jan 26 at 12:53
$begingroup$
This finite field is usually denoted by $Bbb F_p$.
$endgroup$
– Dietrich Burde
Jan 26 at 12:53
$begingroup$
@DietrichBurde;i meant that only
$endgroup$
– user596656
Jan 26 at 12:53
$begingroup$
@DietrichBurde;i meant that only
$endgroup$
– user596656
Jan 26 at 12:53
$begingroup$
Possible duplicate of Irreducible cyclotomic polynomial over a field $mathbb{F}_q$?
$endgroup$
– Dietrich Burde
Jan 26 at 12:54
$begingroup$
Possible duplicate of Irreducible cyclotomic polynomial over a field $mathbb{F}_q$?
$endgroup$
– Dietrich Burde
Jan 26 at 12:54
$begingroup$
You can use $mathbb{Z}_p$ for the cyclic group but in Galois field theory $mathbb{Z}_p$ always refers to the $p$-adic integers (it is not a field but your question still made sense there)
$endgroup$
– reuns
Jan 26 at 12:58
$begingroup$
You can use $mathbb{Z}_p$ for the cyclic group but in Galois field theory $mathbb{Z}_p$ always refers to the $p$-adic integers (it is not a field but your question still made sense there)
$endgroup$
– reuns
Jan 26 at 12:58
|
show 6 more comments
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$begingroup$
@DietrichBurde;$Bbb Z_p$ means the ring $Bbb Z_n$ where $n=p$ and it is a field
$endgroup$
– user596656
Jan 26 at 12:51
$begingroup$
This finite field is usually denoted by $Bbb F_p$.
$endgroup$
– Dietrich Burde
Jan 26 at 12:53
$begingroup$
@DietrichBurde;i meant that only
$endgroup$
– user596656
Jan 26 at 12:53
$begingroup$
Possible duplicate of Irreducible cyclotomic polynomial over a field $mathbb{F}_q$?
$endgroup$
– Dietrich Burde
Jan 26 at 12:54
$begingroup$
You can use $mathbb{Z}_p$ for the cyclic group but in Galois field theory $mathbb{Z}_p$ always refers to the $p$-adic integers (it is not a field but your question still made sense there)
$endgroup$
– reuns
Jan 26 at 12:58