I'm trying to prove the statement:

Any two distinct irreducible monic polynomials are relatively prime.

**Attempt:** Let $\phi_1(x)$ and $\phi_2(x)$ be two distinct irreducible polynomials. Assume they are not relatively prime. Then there exists a polynomial, $h(x)$, such that

$\phi_1(x) = h(x)h'(x) \quad \text{and} \quad \phi_2(x) = h(x)h''(x) $,

for polynomials $h'(x)$ and $h''(x)$ of appropriate degree. But this implies that $\phi_1(x)$ and $\phi_2(x)$ can be expressed as a product of two polynomials. Hence, they can't be irreducible.

**Comment:** I'm not too sure if the proof is correct. I haven't seemed to have used the conditions that the polynomials are both distinct and monic. What am I doing wrong?

I don't have much background in abstract algebra. I came across this thereom in an appendix on polynomials in a book on liner algebra, so I am going through it to cover the unit on minimal polynomials.