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I wonder if the following is known or, not very difficult to see:

Let $K$ be a number field and $A, B \in \mathcal{O}_K$ be nonzero integers of $K$. Does there necessarily exist a positive integer $n > 1$ such that the polynomial $x^n + Ax + B$ were irreducible over $K[x]$?

Thank you, Albertas

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    $\begingroup$ Not if $1+A+B=0$, since then it is divisible by $x-1$. $\endgroup$ Aug 8, 2013 at 19:09
  • $\begingroup$ 2David: Thank you for the answer to the question as stated. 2Igor: Thank you, I am aware of that paper. While this question may still possibly be of general interest for generic $A, B$, I realized that I mis-stated the problem that was of interest to me, which is in fact irreducibility of $x^n + x + \alpha^n$ for some $n$ when $\alpha \in K$ is fixed. $\endgroup$
    – Albertas
    Aug 9, 2013 at 18:00

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The question of irreducibility of trinomial has been studied: see Selmer's 1956 math. skand paper. According to Selmer, the only criterion he was aware of (in 1956) for the irreducibility of a general trinomial is due to Nagell and says:

$x^n + q x^p + r$ is irreducible if $|q| > 1+ |r|^{n-1}$ AND if $h>1$ divides $n,$ then $|r|$ is not an $h$-th power.

In the special case that $r = \pm 1, p=1,$ the polynomial is irreducible whenever $|a| \geq 3,$ (this is due to Perron), and Selmer analyzes the remaining case.

Selmer's paper is available for free from www.mscand.dk.

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    $\begingroup$ I believe Selmer's result is only over the rationals, and in fact his method does not extend to number fields. $\endgroup$ Aug 8, 2013 at 23:25

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