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How can we find number of primitive elements in the field $\mathbb{F}_9$.

$\mathbb{Z}_3[x]/(x^2+1) = \mathbb{F}_9$ and $α^4=1$. I could not improve the solution. I missunderstand the conception. can you help me.

thanks.

tc1729
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spectralmath
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  • Do you know group theory in general? – Tobias Kildetoft Oct 21 '13 at 18:07
  • i know but i cant compare in finite fields. i do not know how to think. if i think simply i get no primitive elements in F9 because every element has order less than 8 which contradicts with the ord(primitive element) – spectralmath Oct 21 '13 at 18:11
  • What is you $a$? And you did not really answer my question about group theory? Do you know basic group theory (cyclic group and such?) – Tobias Kildetoft Oct 21 '13 at 18:15
  • α is root. i answered. i said i knew. – spectralmath Oct 21 '13 at 18:17
  • What do you mean you can't compare in finite fields? You are asked about the number of generators of the multiplicative group of the field, which is a cyclic group of order $4$. – Tobias Kildetoft Oct 21 '13 at 18:19
  • original question is number of primitive elements in F9(Field). those are just my predictions about solution. – spectralmath Oct 21 '13 at 18:20
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    By the calculation in your previous question, $\varphi(8)$. – André Nicolas Oct 21 '13 at 18:21
  • what is generalization of this. can we do this for all numbers or except prime. for instance F25 and F32.. because i know for F19 we have {2,3,10,13,14,15} for primitive elements. – spectralmath Oct 21 '13 at 18:23
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    One primitive element is $1+\alpha$. There are three more. – Derek Holt Oct 21 '13 at 19:39
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    No point in checking the elements of the subfield $\Bbb{F}_3$. As you observed $\alpha$ has order four. That leaves elements of the form $a+b\alpha$ with $a,b$ both non-zero. Have you checked the order of the element $1+\alpha$ suggested by Derek Holt? – Jyrki Lahtonen Nov 05 '13 at 15:35
  • 1+α is primitive because the order of 1+α is 8. (1+α)^4=2. then 2^2=1. F9={0,1,2,α,α+1,α+2,2α,2α+1,2α+2} number of Primitive elements: φ(9-1)=φ(8)=2.(3-1)=4 So, I checked one by one. that has order 8. Primitive elements are: {α+1,α+2,2α+1,2α+2} is that right? @JyrkiLahtonen – spectralmath Nov 08 '13 at 09:37
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    Correct. Why don't you write that as an answer!? You get some upvotes, comments, and improve the site hygiene by removing your question from the unanswered queue :-) – Jyrki Lahtonen Nov 08 '13 at 09:54
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    i am new in here and i newly get that if i do this, it is beneficial for humanity =) thank you very much for your help. – spectralmath Nov 08 '13 at 10:18

1 Answers1

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F9={0,1,2,α,α+1,α+2,2α,2α+1,2α+2}

Number of Primitive elements: φ(9-1)=φ(8)=2.(3-1)=4

So, I checked one by one. that has order 8.

Primitive elements are: {α+1,α+2,2α+1,2α+2}

{example: 1+α is primitive because the order of 1+α is 8. (1+α)^4=2. then 2^2=1.}

solved by Assoc.Prof @Jyrki Lahtonen and other mathematician 's helps and comments.

spectralmath
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