Any tips on how to write the multiplication table in general for a finite field of polynomials (specific example: $F = (\mathbb{Z}/2\mathbb{Z})[x]/(x^2+x+1)$. I know that $F$ has four elements here, $\{0,1,x,x+1\}$, and I know how to make a multiplication table for this. But it gets very tricky with "bigger" fields of polynomials, as modding elements becomes difficult. Any tips?
Answer
If you're in one variable modding elements is just long division. It can be a little tedious but it is a straightforward process. If things are really big and you don't want to do it by hand there are free computer programs that can handle this. You can also try to use wofphram alpha. If it gives you an answer over the rationals then you can clear fractions and reduce mod $p$ to get an answer over a prime field.
If you'd like to do it by hand having a normal form for representatives of the cosets is helpful. It's easier to tell when things are equal that way. For the example above the normal form would be a coset representative that has only a constant and degree $1$ term (because you know $x^2 = x + 1$ in the quotient). Having a normal form will also work in a lot of situations where multiple variables are involved.
If there are multiple variables and no obvious normal form then you can resort to a Grobner basis. You really don't want to do this way by hand though. Again, computer programs are helpful here. I would recommend Macaulay2.
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