I'm studing polynomials; I have this exercise:
Find the irreducible factors of the polynomial $x^4-2x^2-3 \in \mathbb{Z}_5[x]$
I think in this way: I need to find root of the polynomial. A root of polynomial is a number such that the polynomials application $f(x) = x^4-2x^2-3=0$ that means $x^4-2x^2-3 \equiv 0 \pmod 5$, so $5|x^4-2x^2-3$. The only $x$ that makes this possible is $x=2$ (in fact $5|5$) and $x=3$ (in fact $5|60$). I know that this is a correct way, but how can I find roots if I'm in $\mathbb{Z}_{430}$, obviously I can't try this 430 times. Again: what if I'm in $\mathbb{R}[x]$?
Anyway, the next step is to divide the polynomial by $x-i$ where $i$ are my roots. So
$\frac{x^4-2x^2-3}{x-2}= x^3+2x^2+2x+4$ $\frac{x^3+2x^2+2x+4}{x-3} = x^2+2$
Since $x^2+2$ is irreducible, the factorization is $(x-2)(x-3)(x^2+2)$ is right?