How do I go about solving exercises such as this one:
Find all polynomials $f(x)$ in $\mathbb{Z}_3$ that satisfy
$$f(x) \equiv 1 \space \space \mathrm{mod} \space \space x^2 + 1$$ $$f(x) \equiv x \space \space \mathrm{mod} \space \space x^3 + 2x + 2$$
in $\mathbb{Z}_3.$
I know about the Chinese Remainder Theorem, but only how to apply it to system of congruences where there are no polynomials involved.
I realise that $f_1(x) \equiv f_2(x) \space \space \mathrm{mod} \space g(x)$ means that $f_1(x) - f_2(x)$ is divisible by $g(x)$, but that's about as far as I've come with this problem.
Also, if anyone has any advice as to where I can read about modular arithmetic involving polynomials, I'd be happy to hear about it, because the literature I have doesn't say much about it at all, and I would like to learn.
Follow the constructive proof of the Chinese Remainder Theorem. Here is a roadmap:
Let $p(x)=x^2+1$ and $q(x)=x^3+2x+2$.
Use the extended Euclidean Algorithm in $\mathbb{Q}[x]$ to find polynomials $u(x)$ and $v(x)$ such that $u(x)p(x)+v(x)q(x)=1$.
Set $f(x)=(1)v(x)q(x)+(x) u(x)p(x)$ mod $3$.