$x^{2n} + x^{2n-1} + x^ {2n-2} +ldots+ x + 1$ is irreducible for any $nin mathbb N$ in $F_2[x]$. True or...












1












$begingroup$


Will the polynomials of the following set $A$ be irreducible in $F_2[x]$?



$A = [x^{2n} + x^{2n-1} + x^ {2n-2} + ldots+ x + 1 : nin mathbb N]$



Can anyone please give me hints how to proceed?



Every term is there. I meant every polynomial of degree $2n$ has $2n+1$ terms.






abstract-algebra polynomials ring-theory irreducible-polynomials







share|cite|improve this question











$endgroup$












  • $begingroup$
    Are you intending to omit any terms among the $x^i$ and if yes then which?
    $endgroup$
    – Sorin Tirc
    Dec 14 '18 at 12:06










  • $begingroup$
    No no every term is there. I mean every polynomial of degree $2n$ has $2n+1$ terms.@SorinTirc
    $endgroup$
    – cmi
    Dec 14 '18 at 12:17






  • 1




    $begingroup$
    I am asking this because you omitted $x^{2n-2}$ so maybe you should correct that and then my answer below should be an answer to your question
    $endgroup$
    – Sorin Tirc
    Dec 14 '18 at 12:21
















1












$begingroup$


Will the polynomials of the following set $A$ be irreducible in $F_2[x]$?



$A = [x^{2n} + x^{2n-1} + x^ {2n-2} + ldots+ x + 1 : nin mathbb N]$



Can anyone please give me hints how to proceed?



Every term is there. I meant every polynomial of degree $2n$ has $2n+1$ terms.






abstract-algebra polynomials ring-theory irreducible-polynomials







share|cite|improve this question











$endgroup$












  • $begingroup$
    Are you intending to omit any terms among the $x^i$ and if yes then which?
    $endgroup$
    – Sorin Tirc
    Dec 14 '18 at 12:06










  • $begingroup$
    No no every term is there. I mean every polynomial of degree $2n$ has $2n+1$ terms.@SorinTirc
    $endgroup$
    – cmi
    Dec 14 '18 at 12:17






  • 1




    $begingroup$
    I am asking this because you omitted $x^{2n-2}$ so maybe you should correct that and then my answer below should be an answer to your question
    $endgroup$
    – Sorin Tirc
    Dec 14 '18 at 12:21














1












1








1


1



$begingroup$


Will the polynomials of the following set $A$ be irreducible in $F_2[x]$?



$A = [x^{2n} + x^{2n-1} + x^ {2n-2} + ldots+ x + 1 : nin mathbb N]$



Can anyone please give me hints how to proceed?



Every term is there. I meant every polynomial of degree $2n$ has $2n+1$ terms.






abstract-algebra polynomials ring-theory irreducible-polynomials







share|cite|improve this question











$endgroup$




Will the polynomials of the following set $A$ be irreducible in $F_2[x]$?



$A = [x^{2n} + x^{2n-1} + x^ {2n-2} + ldots+ x + 1 : nin mathbb N]$



Can anyone please give me hints how to proceed?



Every term is there. I meant every polynomial of degree $2n$ has $2n+1$ terms.






abstract-algebra polynomials ring-theory irreducible-polynomials




abstract-algebra polynomials ring-theory irreducible-polynomials






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Dec 14 '18 at 15:09







cmi

















asked Dec 14 '18 at 11:58









cmicmi

1,141312




1,141312












  • $begingroup$
    Are you intending to omit any terms among the $x^i$ and if yes then which?
    $endgroup$
    – Sorin Tirc
    Dec 14 '18 at 12:06










  • $begingroup$
    No no every term is there. I mean every polynomial of degree $2n$ has $2n+1$ terms.@SorinTirc
    $endgroup$
    – cmi
    Dec 14 '18 at 12:17






  • 1




    $begingroup$
    I am asking this because you omitted $x^{2n-2}$ so maybe you should correct that and then my answer below should be an answer to your question
    $endgroup$
    – Sorin Tirc
    Dec 14 '18 at 12:21


















  • $begingroup$
    Are you intending to omit any terms among the $x^i$ and if yes then which?
    $endgroup$
    – Sorin Tirc
    Dec 14 '18 at 12:06










  • $begingroup$
    No no every term is there. I mean every polynomial of degree $2n$ has $2n+1$ terms.@SorinTirc
    $endgroup$
    – cmi
    Dec 14 '18 at 12:17






  • 1




    $begingroup$
    I am asking this because you omitted $x^{2n-2}$ so maybe you should correct that and then my answer below should be an answer to your question
    $endgroup$
    – Sorin Tirc
    Dec 14 '18 at 12:21
















$begingroup$
Are you intending to omit any terms among the $x^i$ and if yes then which?
$endgroup$
– Sorin Tirc
Dec 14 '18 at 12:06




$begingroup$
Are you intending to omit any terms among the $x^i$ and if yes then which?
$endgroup$
– Sorin Tirc
Dec 14 '18 at 12:06












$begingroup$
No no every term is there. I mean every polynomial of degree $2n$ has $2n+1$ terms.@SorinTirc
$endgroup$
– cmi
Dec 14 '18 at 12:17




$begingroup$
No no every term is there. I mean every polynomial of degree $2n$ has $2n+1$ terms.@SorinTirc
$endgroup$
– cmi
Dec 14 '18 at 12:17




1




1




$begingroup$
I am asking this because you omitted $x^{2n-2}$ so maybe you should correct that and then my answer below should be an answer to your question
$endgroup$
– Sorin Tirc
Dec 14 '18 at 12:21




$begingroup$
I am asking this because you omitted $x^{2n-2}$ so maybe you should correct that and then my answer below should be an answer to your question
$endgroup$
– Sorin Tirc
Dec 14 '18 at 12:21










1 Answer
1






active

oldest

votes


















2












$begingroup$

It's false in general, for example for $n=4$ : $x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1 = (x^2+x+1)(x^6+x^3+1)$.



In general $frac{x^n-1}{x-1}$ is irreducible iff n is prime(the cyclotomic poly) over $ mathbb{Q}[X]$






share|cite|improve this answer











$endgroup$













  • $begingroup$
    I think it's intended to omit the terms $x^6, x^4, x^2$.
    $endgroup$
    – Slade
    Dec 14 '18 at 12:02










  • $begingroup$
    oh, didn't see that, but then I am not quite sure exactly which terms are omitted tbh
    $endgroup$
    – Sorin Tirc
    Dec 14 '18 at 12:04










  • $begingroup$
    All the even terms are omitted except the first one.
    $endgroup$
    – Slade
    Dec 14 '18 at 12:04










  • $begingroup$
    and the last one $x^0$ then too maybe :) ? In whice case the n=2 poly is reducible :)
    $endgroup$
    – Sorin Tirc
    Dec 14 '18 at 12:05












  • $begingroup$
    Can we say if a polynomial is irreducible in $mathbb Q[x]$ so will be in $F_2[x]$?@SorinTirc
    $endgroup$
    – cmi
    Dec 14 '18 at 15:08












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1 Answer
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1 Answer
1






active

oldest

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active

oldest

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active

oldest

votes









2












$begingroup$

It's false in general, for example for $n=4$ : $x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1 = (x^2+x+1)(x^6+x^3+1)$.



In general $frac{x^n-1}{x-1}$ is irreducible iff n is prime(the cyclotomic poly) over $ mathbb{Q}[X]$






share|cite|improve this answer











$endgroup$













  • $begingroup$
    I think it's intended to omit the terms $x^6, x^4, x^2$.
    $endgroup$
    – Slade
    Dec 14 '18 at 12:02










  • $begingroup$
    oh, didn't see that, but then I am not quite sure exactly which terms are omitted tbh
    $endgroup$
    – Sorin Tirc
    Dec 14 '18 at 12:04










  • $begingroup$
    All the even terms are omitted except the first one.
    $endgroup$
    – Slade
    Dec 14 '18 at 12:04










  • $begingroup$
    and the last one $x^0$ then too maybe :) ? In whice case the n=2 poly is reducible :)
    $endgroup$
    – Sorin Tirc
    Dec 14 '18 at 12:05












  • $begingroup$
    Can we say if a polynomial is irreducible in $mathbb Q[x]$ so will be in $F_2[x]$?@SorinTirc
    $endgroup$
    – cmi
    Dec 14 '18 at 15:08
















2












$begingroup$

It's false in general, for example for $n=4$ : $x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1 = (x^2+x+1)(x^6+x^3+1)$.



In general $frac{x^n-1}{x-1}$ is irreducible iff n is prime(the cyclotomic poly) over $ mathbb{Q}[X]$






share|cite|improve this answer











$endgroup$













  • $begingroup$
    I think it's intended to omit the terms $x^6, x^4, x^2$.
    $endgroup$
    – Slade
    Dec 14 '18 at 12:02










  • $begingroup$
    oh, didn't see that, but then I am not quite sure exactly which terms are omitted tbh
    $endgroup$
    – Sorin Tirc
    Dec 14 '18 at 12:04










  • $begingroup$
    All the even terms are omitted except the first one.
    $endgroup$
    – Slade
    Dec 14 '18 at 12:04










  • $begingroup$
    and the last one $x^0$ then too maybe :) ? In whice case the n=2 poly is reducible :)
    $endgroup$
    – Sorin Tirc
    Dec 14 '18 at 12:05












  • $begingroup$
    Can we say if a polynomial is irreducible in $mathbb Q[x]$ so will be in $F_2[x]$?@SorinTirc
    $endgroup$
    – cmi
    Dec 14 '18 at 15:08














2












2








2





$begingroup$

It's false in general, for example for $n=4$ : $x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1 = (x^2+x+1)(x^6+x^3+1)$.



In general $frac{x^n-1}{x-1}$ is irreducible iff n is prime(the cyclotomic poly) over $ mathbb{Q}[X]$






share|cite|improve this answer











$endgroup$



It's false in general, for example for $n=4$ : $x^8+x^7+x^6+x^5+x^4+x^3+x^2+x+1 = (x^2+x+1)(x^6+x^3+1)$.



In general $frac{x^n-1}{x-1}$ is irreducible iff n is prime(the cyclotomic poly) over $ mathbb{Q}[X]$







share|cite|improve this answer














share|cite|improve this answer



share|cite|improve this answer








edited Dec 14 '18 at 12:02

























answered Dec 14 '18 at 12:02









Sorin TircSorin Tirc

1,875213




1,875213












  • $begingroup$
    I think it's intended to omit the terms $x^6, x^4, x^2$.
    $endgroup$
    – Slade
    Dec 14 '18 at 12:02










  • $begingroup$
    oh, didn't see that, but then I am not quite sure exactly which terms are omitted tbh
    $endgroup$
    – Sorin Tirc
    Dec 14 '18 at 12:04










  • $begingroup$
    All the even terms are omitted except the first one.
    $endgroup$
    – Slade
    Dec 14 '18 at 12:04










  • $begingroup$
    and the last one $x^0$ then too maybe :) ? In whice case the n=2 poly is reducible :)
    $endgroup$
    – Sorin Tirc
    Dec 14 '18 at 12:05












  • $begingroup$
    Can we say if a polynomial is irreducible in $mathbb Q[x]$ so will be in $F_2[x]$?@SorinTirc
    $endgroup$
    – cmi
    Dec 14 '18 at 15:08


















  • $begingroup$
    I think it's intended to omit the terms $x^6, x^4, x^2$.
    $endgroup$
    – Slade
    Dec 14 '18 at 12:02










  • $begingroup$
    oh, didn't see that, but then I am not quite sure exactly which terms are omitted tbh
    $endgroup$
    – Sorin Tirc
    Dec 14 '18 at 12:04










  • $begingroup$
    All the even terms are omitted except the first one.
    $endgroup$
    – Slade
    Dec 14 '18 at 12:04










  • $begingroup$
    and the last one $x^0$ then too maybe :) ? In whice case the n=2 poly is reducible :)
    $endgroup$
    – Sorin Tirc
    Dec 14 '18 at 12:05












  • $begingroup$
    Can we say if a polynomial is irreducible in $mathbb Q[x]$ so will be in $F_2[x]$?@SorinTirc
    $endgroup$
    – cmi
    Dec 14 '18 at 15:08
















$begingroup$
I think it's intended to omit the terms $x^6, x^4, x^2$.
$endgroup$
– Slade
Dec 14 '18 at 12:02




$begingroup$
I think it's intended to omit the terms $x^6, x^4, x^2$.
$endgroup$
– Slade
Dec 14 '18 at 12:02












$begingroup$
oh, didn't see that, but then I am not quite sure exactly which terms are omitted tbh
$endgroup$
– Sorin Tirc
Dec 14 '18 at 12:04




$begingroup$
oh, didn't see that, but then I am not quite sure exactly which terms are omitted tbh
$endgroup$
– Sorin Tirc
Dec 14 '18 at 12:04












$begingroup$
All the even terms are omitted except the first one.
$endgroup$
– Slade
Dec 14 '18 at 12:04




$begingroup$
All the even terms are omitted except the first one.
$endgroup$
– Slade
Dec 14 '18 at 12:04












$begingroup$
and the last one $x^0$ then too maybe :) ? In whice case the n=2 poly is reducible :)
$endgroup$
– Sorin Tirc
Dec 14 '18 at 12:05






$begingroup$
and the last one $x^0$ then too maybe :) ? In whice case the n=2 poly is reducible :)
$endgroup$
– Sorin Tirc
Dec 14 '18 at 12:05














$begingroup$
Can we say if a polynomial is irreducible in $mathbb Q[x]$ so will be in $F_2[x]$?@SorinTirc
$endgroup$
– cmi
Dec 14 '18 at 15:08




$begingroup$
Can we say if a polynomial is irreducible in $mathbb Q[x]$ so will be in $F_2[x]$?@SorinTirc
$endgroup$
– cmi
Dec 14 '18 at 15:08


















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