So I'm trying to prove this Proposition:
Every polynomial of degree greater than $0$ is divisible by an irreducible polynomial.
Consider a polynomial $P \in \mathbb R [x]$ of degree $\geq 1$
Let $S$ denote the set of non-trivial factors of $P$.
Suppose that $S$ does not contain any irreducible polynomials.
By the well ordering principle let $P_0$ denote the polynomial of least degree in $S$
By our assumption $P_0$ is not irreducible $\therefore P_0=P_1P_2$ for some
$P_1,P_2 \in \mathbb R[x]$
This contradicts the definition of $P_0$ being the polynomial of least degree to be a factor of $P$
Thus we are forced to conclude the polynomial of least degree is irreducible and the claim follows.
My main issue is can I use the well ordering principle in this fashion?