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Math Help - polynomial rings

  1. #1
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    polynomial rings

    \mathbb{Z}_3[x]/(x^2+1) is a field of order 9. How do I find a generator for the cyclic multiplicative group of nonzero elements?
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    Quote Originally Posted by dori1123 View Post
    \mathbb{Z}_3[x]/(x^2+1) is a field of order 9. How do I find a generator for the cyclic multiplicative group of nonzero elements?
    Let \alpha = x + (x^2+1). All elements in this field are a+b\alpha where a,b\in \mathbb{Z}_3. Therefore, the elements are:
    0, \alpha, 2\alpha, 1, 1+\alpha, 1+2\alpha, 2, 2+\alpha, 2+2\alpha.

    You also know that \alpha^2 + 1 = 0 so \alpha^2 = 2.
    From this list find an element y so that y^4\not = 1.
    And that would be your generator.
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    Why are you letting \alpha = x + (x^2+1), not \alpha=x^2+1.
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    Quote Originally Posted by dori1123 View Post
    Why are you letting \alpha = x + (x^2+1), not \alpha=x^2+1.
    First because \alpha = x^2+1 makes no sense since \alpha needs to be an element of \mathbb{Z}_3[x]/(x^2+1) and \mathbb{Z}_3[x]/(x^2+1) consists of cosets of (x^2+1). Second, \{1 ,\alpha\} is a basis for this field over \mathbb{Z}_3. Therefore, everything has the form a+b\alpha.
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