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Tell me please, if there is a field that contains 121 elements and is an element of order 12 in the multiplicative group of this field, how can I find the minimal polynomial of this element?

GThompson
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    Welcome to stackexchange. You are more likely to get help instead of downvotes and votes to close if you edit your question to tell us what you understand about the problem, what you have tried and where you are stuck. – Ethan Bolker May 09 '18 at 12:02
  • Tell me please, minimal polynomial over $\mathbb{F}_{11}$? – Dietrich Burde May 09 '18 at 12:03
  • @Dietrich Burde, i think yes. – GThompson May 09 '18 at 12:09
  • if I correctly understand that the multiplicative field group is isomorphic to Z /120Z and, for example, element 8 is of order 12. But how can I find the minimal polynomial of this element? – GThompson May 09 '18 at 12:11
  • See here, how it is done; just replace $2$ by $11$... – Dietrich Burde May 09 '18 at 12:12
  • Thank you so much! – GThompson May 09 '18 at 12:15
  • Or note that it has degree $2$ and must divide the $12$th cyclotomic polynomial $x^4-x^2+1$, so you just need to factorize that poynomial into two quadratic factors over ${\mathbb F}_{11}$. The two factors are both possible answers. – Derek Holt May 09 '18 at 12:34
  • @DerekHolt, sorry, but if I needed a minimal polynomial of an element of order 15, would I be looking for the decomposition of 15th cyclotomic polynomial into two, which will be possible answers? – GThompson May 09 '18 at 12:59
  • In that case it would be easier to factorize $x^3-4$, for example, because $4$ has order $5$. – Derek Holt May 09 '18 at 13:18

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