3

Is there a relation between the irreducibility of a polynomial and its derivative under certain conditions?

Michael Hardy
  • 1
  • Are you asking: when is the derivative of an irreducible polynomial itself irreducible? – lhf May 30 '16 at 19:36
  • Do you have some particular conditions in mind? – Eric Wofsey May 30 '16 at 19:37
  • Are you talking about polynomials over $\mathbb{R}$ or over any field/ring ? (So you're talking about formal derivative) – H. Potter May 30 '16 at 20:00
  • My question is not specific because I am not familiar with the topic and I have been unable to come up with things myself :(. I am interested in any result that links irreduciblity and the derivative (formal one over a general field). –  May 30 '16 at 20:22

1 Answers1

1

Very partial answer:

Any polynomial $g(X)$ over the rationals $\mathbb Q$ (or more generally a field of characteristic $0$) with degree $\ge 1$ has antiderivatives $f(X)$ that are reducible. In fact, given one antiderivative $f_0(X)$ you could take $f_0(X) - f_0(c)$ for any $c \in \mathbb Q$, which is divisible by $X-c$.

So having $f(X)$ reducible doesn't tell you anything about whether $f'(X)$ is irreducible.

Of course, if $f(X)$ is divisible by $q(X)^2$ for some polynomial $q$ of degree $> 1$, then $f'(X)$ is divisible by $q(X)$. If the degree of $f(X)$ is at least $3$, $f'(X)$ is then reducible.

Robert Israel
  • 448,999
  • I understand that there won't be a magical link between them (since if you look at the anti derivative, if it comes out reducible you can add $c$ so that it's irreducible (one of my previous question on this site)). I'm hoping that under some conditions lies a link.

    The reason I suspect that under suitable conditions this might be possible is because testing on wolfram alpha, the taylor expansion of e^x up to a certain degree seems to be irreducible.

     –  May 30 '16 at 20:27
  • 1
    Well, that would be a rather specialized result, and you should ask about it in a separate question. You're unlikely to get a response from this very general question that sheds light on it. – Robert Israel May 30 '16 at 21:27
  • But I'd love a generalized result :), in any case I'll take your suggestion and post the specific question later. –  May 31 '16 at 05:31