What are irreducible factors?

Question

What are Irreducible factors? I have to solve this question:

Find the irreducible factors of $x^4 + 5x^3 + 8x^2 + 9x + 10$ in ${\bf Z}_{11} [x]$.

I couldn't find any websites that explained this clearly and our course notes aren't that helpful. I'm pretty confused so any help would be great.

Answer 1

Note that $1 \in \mathbb{Z}_{11}$ is a root of this polynomial $f(x)$, hence $(x-1)\mid f(x)$. Check that $$ g(x) := \frac{f(x)}{(x-1)} = x^3 +6x^2 +3x +1 $$ Also, $1$ is a root of $g(x)$, so $(x-1)\mid g(x)$. And $$ h(x) := \frac{g(x)}{(x-1)} = x^2 +7x + 10 $$ Now can you check if this factors?

Answer 2

Irreducible factors are like prime numbers, but for polynomials. You can't write them as a product of lower order polynomials.

Example : $x^2 - 1$ is not irreducible in $\mathbb Z[x]$ as it can be written as $(x-1)(x+1)$
$x^2 + 1$ is irreducible in $\mathbb Z[x]$ as it cannot be written as a product of lower order polynomials.
However, $x^2 + 1$ is reducible in $\mathbb Z_2[x]$ as $x^2+1 \equiv x^2 - 1 = (x+1)(x-1)$

You can find the definition of irreducible here. An excellent theorem to check irreducibility in $\mathbb Z[x]$ is the Eisenstein's Criterion. There are other ways to check of course.