Let $K|k$ be a extension field (i.e, $K|k$ denotes that $k \subseteq K$, where both $K,k$ are fields) with $K$ the descomposition field of $p(x)\in k[x]$.
I don't get the following:
If in the descomposition of $p(x)$ in $k[x]$ there is a irreducible $r(x)$ such that $\text{deg}(r(x))\geq 2$, then exists $\alpha \in K$ and $r(\alpha)=0$.
Sorry if this is too basic.
$p(x)$ might be the product of irreducible polynomials of degrees $\geq 2$ in $k[x]$ but in $K[x]$, it is the product of linear factors. Thus, each irreducible divisor $r(x)$ of $p(x) \in k[x]$ also splits into linear factors in $K[x]$. So there is an $\alpha$ in $K$ such that $r(\alpha) = 0$ since $r(x) = \prod_{i} (x-\alpha_i), \alpha_i \in K$.