Irreducibility of the polynomial $x^n+5x+3$ over $\mathbb{Q}$

Asked Mar 17 '24 at 04:45

Active Mar 22 '24 at 05:42

Viewed 232 times

(Now asked on MO: https://mathoverflow.net/questions/467368 )

I would like to determine whether the polynomial $x^n+5x+3$ is irreducible over $\mathbb{Q}$ for $n\ge 2$. I thought of considering $\bmod{2}$ so that we obtain a polynomial $x^n+x+1$. However the polynomial $x^n+x+1$ is not always irreducible over $\mathbb{F}_2$ (see here) and I was stuck.

I also used matlab to check the irreducibility of such a polynomial. The result says for $2\le n\le 100$, $x^n+5x+3$ is always irreducible.

Can you help me with this problem? Thanks in advance.

edited Mar 22 '24 at 05:42

Sil

16,612

asked Mar 17 '24 at 04:45

MathLearner

2

Have you tried shifted eisentein? – nozalp10 Mar 17 '24 at 07:07
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You can also try one of the ideas given in this post, for $X^n+5X^{n-1}+3$. It is very similar. – Dietrich Burde Mar 17 '24 at 10:41
Also this post. – Dietrich Burde Mar 17 '24 at 10:47
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@DietrichBurde The problems looks similar, but it seems that neither the answers in those posts nor shifted Eisenstein can be applied to this problem. – Benjamin Wright Mar 18 '24 at 22:24
Standard criteria do not seem to apply to this one, if you don't find your answer here you might try your luck on MO. – Sil Mar 20 '24 at 07:49
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It is easy to see that the polynomial is irreducible modulo five whenever $n$ is a power of two. While very far from a solution at least we know it to be irreducible for infinitely many choices of $n$ :-) – Jyrki Lahtonen Mar 22 '24 at 06:15

Irreducibility of the polynomial $x^n+5x+3$ over $\mathbb{Q}$

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