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.

 

[tacman]

Hi Rayne,

Thank you for reaching out and sharing your concerns about polynomial algebra in Z_3[x] modulo 1+x^2. I understand that it can be confusing at first, but with some practice and understanding of the concept, you will be able to solve these problems easily.

To answer your question, the members of F[x] are indeed permutations of polynomials of degree no higher than 2. In other words, they are polynomials with coefficients in Z_3, but with a restriction that the highest degree term must be x^2 or lower. So, examples of members of F[x] could be 1, x, 2x^2+1, 2x+2, etc.

To compute the addition table, we first need to define the addition operation in F[x]. As mentioned in the problem, addition and multiplication are defined in the usual way, but modulo 1+x^2. This means that we perform addition and multiplication as we would normally do with polynomials, but then we take the remainder when divided by 1+x^2. So, for example, (2x+1) + (x^2+2) would be (2x+1) + (x^2+2) = (x^2+2x+3) which, when divided by 1+x^2, gives us a remainder of 2x+1. This is the result we would put in the addition table for the entry corresponding to (2x+1) and (x^2+2).

To fill in the table, you can indeed consider all possible "combinations" of polynomials and perform the addition operation as described above. Remember to keep the coefficients in Z_3 and take the remainder when divided by 1+x^2. I suggest starting with simpler polynomials and then gradually moving on to more complex ones to get a better understanding.

I hope this helps clear your doubts about solving polynomial algebra in Z_3[x] modulo 1+x^2. Don't hesitate to ask if you have any further questions. Good luck with your studies!

Best regards,