We have the field $GF(3)[x]_{x^2+1}$ and want to multiply the two polynomials $2x+1$ and $2x+2$.
I understand that $GF(3)[x]_{x^2+1} := \{0,1,2,x,2x,x+1,x+2,2x+1,2x+2\}$. I'm not sure what the best method for multiplying polynomials is (without the table).
I first proceed like so: $(2x+1) * (2x+2) = 4x^2+4x+2x+2 = x^2+2$
Now the polynomial can't have a degree bigger than 1 in our field, so I'd have to devide the result I'm getting by $x^2+1$.
$x^2+2 = (x^2+1) + 1$ so we have a remainder of $1$. Is the final result $= 1$? or am I doing the computation wrong from the get go?
In general, the result has to be an element of our field. So do you just take the remainder as the final result?
Thanks in advance