
Polynomial Algebra
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,...,p1}.
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 1row and xcolumn 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

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I have no idea what you mean. I will assume you mean "modulo" as in factor rings.