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Thread: Finding Roots of Polynomials over Polynomial Quotient Rings

  1. #1
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    Finding Roots of Polynomials over Polynomial Quotient Rings

    Hi all,

    Just wondering how I would go about finding roots of an polynomial,
    say t^3 +t^2+1 or t^3+1, over a quotient ring, say Z/2Z[x] / <x^3+x+1>. Is there a general way to do this or do I have to plug and chug all the elements of the field? This is annoying for fields with more than 4 elements! I've programmed the computer solve them for me (plug and chug way) but I don't know how to do it without the computer.


    Thanks,

    Julian
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  2. #2
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    Quote Originally Posted by aznmaven View Post
    Hi all,

    Just wondering how I would go about finding roots of an polynomial,
    say t^3 +t^2+1 or t^3+1, over a quotient ring, say Z/2Z[x] / <x^3+x+1>. Is there a general way to do this or do I have to plug and chug all the elements of the field? This is annoying for fields with more than 4 elements! I've programmed the computer solve them for me (plug and chug way) but I don't know how to do it without the computer.
    Let $\displaystyle K = \mathbb{Z}_2[x]/(x^3+x+1)$.

    Let $\displaystyle \alpha = x + (x^3 + x + 1)$. Notice that $\displaystyle \alpha^3 + \alpha + 1 = 0$. Therefore, $\displaystyle \alpha^3 = \alpha + 1$ - remember $\displaystyle \text{char}(K)=2$.

    You want to determine if $\displaystyle t^3+t^2+1$ has any roots. A messy approach here is to notice that any element in $\displaystyle K$ can be uniquely written as $\displaystyle a+b\alpha + c\alpha^2$ where $\displaystyle a,b,c\in \mathbb{Z}_2$. Substitute that into the polynomial and equate coefficient to zero, and remember that $\displaystyle \alpha^3 = \alpha + 1$.
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