Let $F$ be a field, $f \in F[x]$ of degree 2 or 3. Theorem: If $f$ has no roots, then $f$ is irreducible.
I have shown in several examples the above theorem holds. I am trying to find an example where the theorem does not hold in a degree-4 polynomial in $\mathbb{Q}[x]$.
The quartic polynomial that I have tried was $(x^4 - 22x^2 + 1)$. By the Rational Roots Theorem, the only possible rational roots for $(x^4 - 22x^2 + 1)$ are $\pm$1. But $(1)^4 - 22(1)^2 + 1 \neq 0$ and $(-1)^4 - 22(-1)^2 + 1 \neq 0$. Hence $(x^4 - 22x^2 + 1)$ has no roots. As I'm trying to show that $(x^4 - 22x^2 + 1)$ is reducible, I know that (from WolfRamAlpha) "A polynomial is said to be irreducible if it cannot be factored into nontrivial polynomials over the same field." So, if I can't factor $(x^4 - 22x^2 + 1)$ into nontrivial polynomials over $\mathbb{Q}[x]$, then $(x^4 - 22x^2 + 1)$ over $\mathbb{Q}[x]$ is irreducible. Since I don't know how to factor $(x^4 - 22x^2 + 1)$ into nontrivial polynomials over $\mathbb{Q}[x]$, can I conclude that $(x^4 - 22x^2 + 1)$ over $\mathbb{Q}[x]$ is reducible?