Generator of $mathbb{F}_{q}^*$ where $q$ is not prime.












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$begingroup$


I'm trying to find the generator of the multiplicative group $mathbb{F}_{8}^*$, where $mathbb{F}_8$ is a field. If the order of the field is prime, then it is easy since in that case the field would be isomorphic to the integers modulo, say p. However, for cases where the order of the field is not prime, I'm stuck. Can anyone help me out?



Thanks in advance.






abstract-algebra field-theory







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$endgroup$








  • 3




    $begingroup$
    By $Bbb F_q^x$ do you mean the multiplicative group of nonzero elements of $Bbb F_q$? This is usually denoted as $Bbb F_q^times$ or $Bbb F_q^*$. When $q=8$, it has seven elements, and seven is a prime number, so any element other than the identity is a generator.
    $endgroup$
    – Lord Shark the Unknown
    Jan 1 at 11:56










  • $begingroup$
    @Xenidia Can you please explain your multiplicative group $mathbb{F}^x_q$? Is it the same as $mathbb{F}^*_q$
    $endgroup$
    – toric_actions
    Jan 1 at 12:04












  • $begingroup$
    See also the answers to this question on $Bbb{F}_q^{times}$. In general, see also this duplicate.
    $endgroup$
    – Dietrich Burde
    Jan 1 at 13:56












  • $begingroup$
    So $t + (t^3+t+1)$ is a generator of $mathbb{F}_2[t]/(t^3+t+1)^times cong mathbb{F}_8^times$
    $endgroup$
    – reuns
    Jan 1 at 17:02


















0












$begingroup$


I'm trying to find the generator of the multiplicative group $mathbb{F}_{8}^*$, where $mathbb{F}_8$ is a field. If the order of the field is prime, then it is easy since in that case the field would be isomorphic to the integers modulo, say p. However, for cases where the order of the field is not prime, I'm stuck. Can anyone help me out?



Thanks in advance.






abstract-algebra field-theory







share|cite|improve this question











$endgroup$








  • 3




    $begingroup$
    By $Bbb F_q^x$ do you mean the multiplicative group of nonzero elements of $Bbb F_q$? This is usually denoted as $Bbb F_q^times$ or $Bbb F_q^*$. When $q=8$, it has seven elements, and seven is a prime number, so any element other than the identity is a generator.
    $endgroup$
    – Lord Shark the Unknown
    Jan 1 at 11:56










  • $begingroup$
    @Xenidia Can you please explain your multiplicative group $mathbb{F}^x_q$? Is it the same as $mathbb{F}^*_q$
    $endgroup$
    – toric_actions
    Jan 1 at 12:04












  • $begingroup$
    See also the answers to this question on $Bbb{F}_q^{times}$. In general, see also this duplicate.
    $endgroup$
    – Dietrich Burde
    Jan 1 at 13:56












  • $begingroup$
    So $t + (t^3+t+1)$ is a generator of $mathbb{F}_2[t]/(t^3+t+1)^times cong mathbb{F}_8^times$
    $endgroup$
    – reuns
    Jan 1 at 17:02
















0












0








0





$begingroup$


I'm trying to find the generator of the multiplicative group $mathbb{F}_{8}^*$, where $mathbb{F}_8$ is a field. If the order of the field is prime, then it is easy since in that case the field would be isomorphic to the integers modulo, say p. However, for cases where the order of the field is not prime, I'm stuck. Can anyone help me out?



Thanks in advance.






abstract-algebra field-theory







share|cite|improve this question











$endgroup$




I'm trying to find the generator of the multiplicative group $mathbb{F}_{8}^*$, where $mathbb{F}_8$ is a field. If the order of the field is prime, then it is easy since in that case the field would be isomorphic to the integers modulo, say p. However, for cases where the order of the field is not prime, I'm stuck. Can anyone help me out?



Thanks in advance.






abstract-algebra field-theory




abstract-algebra field-theory






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 1 at 12:25









toric_actions

898




898










asked Jan 1 at 11:54









XenidiaXenidia

1,275729




1,275729








  • 3




    $begingroup$
    By $Bbb F_q^x$ do you mean the multiplicative group of nonzero elements of $Bbb F_q$? This is usually denoted as $Bbb F_q^times$ or $Bbb F_q^*$. When $q=8$, it has seven elements, and seven is a prime number, so any element other than the identity is a generator.
    $endgroup$
    – Lord Shark the Unknown
    Jan 1 at 11:56










  • $begingroup$
    @Xenidia Can you please explain your multiplicative group $mathbb{F}^x_q$? Is it the same as $mathbb{F}^*_q$
    $endgroup$
    – toric_actions
    Jan 1 at 12:04












  • $begingroup$
    See also the answers to this question on $Bbb{F}_q^{times}$. In general, see also this duplicate.
    $endgroup$
    – Dietrich Burde
    Jan 1 at 13:56












  • $begingroup$
    So $t + (t^3+t+1)$ is a generator of $mathbb{F}_2[t]/(t^3+t+1)^times cong mathbb{F}_8^times$
    $endgroup$
    – reuns
    Jan 1 at 17:02
















  • 3




    $begingroup$
    By $Bbb F_q^x$ do you mean the multiplicative group of nonzero elements of $Bbb F_q$? This is usually denoted as $Bbb F_q^times$ or $Bbb F_q^*$. When $q=8$, it has seven elements, and seven is a prime number, so any element other than the identity is a generator.
    $endgroup$
    – Lord Shark the Unknown
    Jan 1 at 11:56










  • $begingroup$
    @Xenidia Can you please explain your multiplicative group $mathbb{F}^x_q$? Is it the same as $mathbb{F}^*_q$
    $endgroup$
    – toric_actions
    Jan 1 at 12:04












  • $begingroup$
    See also the answers to this question on $Bbb{F}_q^{times}$. In general, see also this duplicate.
    $endgroup$
    – Dietrich Burde
    Jan 1 at 13:56












  • $begingroup$
    So $t + (t^3+t+1)$ is a generator of $mathbb{F}_2[t]/(t^3+t+1)^times cong mathbb{F}_8^times$
    $endgroup$
    – reuns
    Jan 1 at 17:02










3




3




$begingroup$
By $Bbb F_q^x$ do you mean the multiplicative group of nonzero elements of $Bbb F_q$? This is usually denoted as $Bbb F_q^times$ or $Bbb F_q^*$. When $q=8$, it has seven elements, and seven is a prime number, so any element other than the identity is a generator.
$endgroup$
– Lord Shark the Unknown
Jan 1 at 11:56




$begingroup$
By $Bbb F_q^x$ do you mean the multiplicative group of nonzero elements of $Bbb F_q$? This is usually denoted as $Bbb F_q^times$ or $Bbb F_q^*$. When $q=8$, it has seven elements, and seven is a prime number, so any element other than the identity is a generator.
$endgroup$
– Lord Shark the Unknown
Jan 1 at 11:56












$begingroup$
@Xenidia Can you please explain your multiplicative group $mathbb{F}^x_q$? Is it the same as $mathbb{F}^*_q$
$endgroup$
– toric_actions
Jan 1 at 12:04






$begingroup$
@Xenidia Can you please explain your multiplicative group $mathbb{F}^x_q$? Is it the same as $mathbb{F}^*_q$
$endgroup$
– toric_actions
Jan 1 at 12:04














$begingroup$
See also the answers to this question on $Bbb{F}_q^{times}$. In general, see also this duplicate.
$endgroup$
– Dietrich Burde
Jan 1 at 13:56






$begingroup$
See also the answers to this question on $Bbb{F}_q^{times}$. In general, see also this duplicate.
$endgroup$
– Dietrich Burde
Jan 1 at 13:56














$begingroup$
So $t + (t^3+t+1)$ is a generator of $mathbb{F}_2[t]/(t^3+t+1)^times cong mathbb{F}_8^times$
$endgroup$
– reuns
Jan 1 at 17:02






$begingroup$
So $t + (t^3+t+1)$ is a generator of $mathbb{F}_2[t]/(t^3+t+1)^times cong mathbb{F}_8^times$
$endgroup$
– reuns
Jan 1 at 17:02












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