# Need help understanding finite fields / modulo for polynomials

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 $\mathbb Z_3[x]$.

• Do you know how to do long division on $$\frac{x^3 + x +1}{x^2 + x +1}$$ in ${\mathbb Z}_3[x]$? i.e. compute quotient and remainder? That's what you need to compute the Extended Euclidean algorithm for polynomials. For example, these pdf notes – user2468 Mar 29 '12 at 0:33
• @J.D. I understand how to do polynomial long division, but am kind of confused on how it differs when in $\mathbb Z_3[x]$ – Stephen Young Mar 29 '12 at 0:41