Irreducible Polynomial over a Field I am studying elementary Field Theory. I have a problem that I have been wrestling with for a bit:

Let $p(x)$ be an irreducible polynomial over a field $F$. Suppose that $p(x)$ divides $f_1(x)....f_n(x)$ in $F[x]$. Prove that $p(x)$ divides $f_i(x)$ for some $i$ in {$1,...,n$}.

I'm assuming we can use induction to prove this but I'm not really sure how to go about it.

Thanks!
The base case of the induction is obvious.

Assume the inductive hypothesis $p\mid f_1f_2\cdots f_n$ implies $p\mid f_i$ for some $i\in\{1,\ldots,n\}$

Now, consider $p\mid f_1f_2\cdots f_nf_{n+1}$. If $p\mid f_1f_2\cdots f_n$, we're done by the inductive hypothesis. Suppose not, then $p\mid f_1f_2\cdots f_{n+1}$ and $p\not\mid f_1f_2\cdots f_n$ implies $p\mid f_{n+1}$ since $p$ is irreducible which completes the inductive step.

Addendum: This is basically a generalization of Euclid's lemma for the Euclidean domain of the ring of polynomials in $x$
You have to prove that if $p$ divides $fg$, then $p$ divides $f$ or $p$ divides $g$, then you can proceed easily by induction. The argument is pretty much the same as for integers, you can look it up in any book.