Irreducible Polynomial over a Field

Question

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!

Answer 1

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 $x$

Answer 2

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.