A polynomial $f\in k[x]$ is irreducible if :
1. $\deg(f)\geq 1$
2. If $f=gh$ then $\deg(g)=0$ or $\deg(h)=0$
First of all, this definition confuses me. For example, Let $f=2x^2+2$ and the field be $\mathbb{R}$. By this definition f is irreducible because $f=2(x^2+1)$ and $\deg(2)=0$. Is this correct? I would have thought $f$ is reducible since it can be divided by two..
Anyway, on with the proof:
If $f$ is reducible then $f=gh$ where $g,h\in k[x]$.
There are two cases, either $\deg(g)=\deg(h)=1$ or $\deg(g)=2 \text{ and } \deg(h)=0$. If the first case is true then this implies $f$ has roots. If the second case is true then either $\deg(g)=0 \text{ or } \deg(h)=0$ which would imply that f is irreducible according to the above definition?
Am I missing something or should the second criterion of irreducibility be:
If$~~f=gh \text{ then }\deg(g)=-\infty \text{ or } \deg(h)=-\infty$
Best Answer
It is much simpler to consider Euclidean division for polynomials: for any $\alpha \in k$, divide $f$ by $x-\alpha$: $$f(x)=q(x)(x-\alpha) +r$$ Furthermore, $r=f(\alpha)$, hence
Thus $f(x) $ has a root if and only if $f(x)$ is divisible by some $x-\alpha$, which implies $f$ is reducible.