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I'm taking a class in finite fields and have not been able to conceptualize how modulo + finite fields works in polynomial space. I understand the basic premises of modular arithmetic, but can't work out how to actually generate a finite field of polynomials.

For example:

Find all $f(x)$ and $g(x)$ in $\mathbb Z_3[x]$: $(x^3 + x +1) f(x) + (x^2 + x +1)g(x) = 1$

I know conceptually how to solve this sort of equation when the coefficients are integers and $f(x), g(x)$ are simple variables, but I don't know how to generate fields in $\mathbb Z_3[x]$ and then how exactly to use them to solve this sort of equation for polynomials once I have their $\mathrm{gcd}$ in $\mathbb Z_3[x]$.

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    Thanks everyone, really helpful stuff - learned more toni$g$ht than in the last two months of lectures.2012-03-29

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There aren't any, since the left side is $0$ at $x=1$.

But one can get a general description of $f(x)$ and $g(x)$ if you replace the right-hand side by, say, $x-1$ (or equivalently $x+2$).