# 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.

• Any book? I'm 100% confident that the novel I am reading does not contain the argument Max needs! Apr 10, 2018 at 21:07
• Do you maybe see a post on here that might help? Thanks!
– Max
Apr 10, 2018 at 21:12
• Any book that deals with this material, of course. Apr 10, 2018 at 21:34
• My answer helps exactly as much as the answer above mine. The answer above mine only does the easy induction I mentioned, and doesn't get to the crux of the issue that I explicitly point out: $p | fg \rightarrow p|f \lor p|g$. Apr 11, 2018 at 18:11
• @xyzzyz: I agree that the answer that has been accepted is little better than yours. I do think it would be a kindness to give the OP more help with the interesting property of prime polynomials that you are appealing to. Apr 11, 2018 at 18:37