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Math Help - Prove Primitive over Z_5

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    Prove Primitive over Z_5

    Prove that polynomial x^2+x+2 is a primitive over Z_5 (The polynomial p(x)=x is the generator of multiplicative group of the field
    Z_5[x]/<x^2+x+2>)

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  2. #2
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    Quote Originally Posted by mandy123 View Post
    Prove that polynomial x^2+x+2 is a primitive over Z_5 (The polynomial p(x)=x is the generator of multiplicative group of the field
    Z_5[x]/<x^2+x+2>)

    Don't have a clue?
    it's very simple: an element of \mathbb{F}=\frac{\mathbb{Z}_5[x]}{<x^2+x+2>} is in the form ax + b \ + <x^2+x+2>, \ a,b \in \mathbb{Z}_5, which for simplicity i'll write it as ax+b. so in \mathbb{F} we have: x^2=-x-2. we want to show

    that x^n \neq 1, \ \text{for} \ n=1,2,3,4,6,12. it's clear for n=1. now we have: x^2=-x-2 \neq 1, \ \ x^3=2-x \neq 1, \ \ x^4 = 3x+2 \neq 1, \ \ x^6 = 2 \neq 1, \ \ x^{12}=4 \neq 1. hence the order of x in \mathbb{F}^{\times} is 24. Q.E.D.
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