Solving Polynomial Algebra in Z_3[x] Modulo 1+x^2

[wu_weidong]

Hi all, I have trouble understanding polynomial algebra.

Let F[x] in Z_3[x] be the set of all polynomials modulo 1+x^2 where Z_3[x] is the set of all polynomials with coefficients in Z_3. Addition and multiplication are defined in the usual way but modulo 1+x^2 and the arithmetic of the coefficients are in Z_3. Compute the addition table.

I don't know what members are in F[x]. Are they 1,x,1+x and x^2, that is, permutations of polynomials of degree no higher than 2? I'm trying to draw similarities between polynomial modulo and integer modulo, where I know for the latter, members of Z_p are {0,1,...,p-1}.

Also, after I know what the members are, do I fill in the table by considering all possible "combinations", i.e. 1+x is obtained from the 1-row and x-column of the table, divide that "combination" by 1+x^2, keeping coefficients in Z_3 then take the remainder as the result?

Thank you.

Regards,
Rayne

 

[matt grime]

You were taught polynomial division at high school, probably, when you did integration. That's all it is: remainder on division by x^2+1. So the elements are 0,1,x,x+1,x^2.
'Combination' is just 'add and take remainder after division by x^2+1'. So 1 added to x^2 is 0.