What happens if $f(x)$ is irreducible but $f(g(x))$ is reducible?

Asked Jan 20 '21 at 13:30

Active Jan 20 '21 at 13:40

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$\mathbb{F}$ is a field. What can we learn if $f(x)$ is irreducible while $f(g(x))$ is reducible for $f,g\in\mathbb{F}[x]$?

It seems likely to be a duplicate, but I've not been able find previous questions on this problem, so I'm here... Any reference is welcome.

Possible outcome, maybe close (as much as I can conjure up):

Then the degree of any irreducible factor of $f(g(x))$ is divisible by $\deg{g}$.

edited Jan 20 '21 at 13:40

asked Jan 20 '21 at 13:30

Juggler

What do you wish to learn? Do you have something in mind? – Jeroen van der Meer Jan 20 '21 at 13:32
Yes, in my possibly unreliable memory, there is some conclusion that we can draw, but it has eluded me.... – Juggler Jan 20 '21 at 13:34
The composition of two irreducible polynomials need not be irreducible. Yet if we take $f(x) = x$ then the irreducibility of $f(g(x))$ is trivially the same as that of $g(x)$. I'm not sure how much can be said. – Jeroen van der Meer Jan 20 '21 at 13:35
Please see my edit. – Juggler Jan 20 '21 at 13:39
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Might this help? https://math.stackexchange.com/questions/233627/fgx-has-a-degree-divisible-by-n – Jeroen van der Meer Jan 20 '21 at 13:41
@JeroenvanderMeer Yes, that’s what I’m looking for, thank you! – Juggler Jan 20 '21 at 13:45

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