Finding Roots of Polynomials over Polynomial Quotient Rings

**aznmaven** · Nov 9th 2008, 04:56 PM

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

**ThePerfectHacker** · Nov 9th 2008, 08:05 PM

Originally Posted by aznmaven

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 $K = \mathbb{Z}_2[x]/(x^3+x+1)$ .

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

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

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