Is it true that he polynomial $frac{x^p- 1}{x-1}$ ($p$ is prime) is irreducible in $mathbb{F}_2[x]$ iff $p$...












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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 $mathbb{F}_2[x]$?










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  • $begingroup$
    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
















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 $mathbb{F}_2[x]$?










share|cite|improve this question











$endgroup$












  • $begingroup$
    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














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 $mathbb{F}_2[x]$?










share|cite|improve this question











$endgroup$





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






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 21:27









Batominovski

33.1k33293




33.1k33293










asked Dec 14 '18 at 15:28









cmicmi

1,120312




1,120312












  • $begingroup$
    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












  • $begingroup$
    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










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