Math Help - 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

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 .

Let . Notice that . Therefore, - remember .

You want to determine if has any roots. A messy approach here is to notice that any element in can be uniquely written as where . Substitute that into the polynomial and equate coefficient to zero, and remember that .