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

Let $K$ be a number field and $\alpha \in K$ be nonzero. Does there necessarily exist a positive integer $n > 1$ such that the polynomial $x^n + x + \alpha^n$ were irreducible over $K[x]$?

Thank you.

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    $\begingroup$ The definitive references for reducibility of trinomials are Schinzel's four papers on this topic. You can look there to see what's known. $\endgroup$
    – Michael Zieve
    Commented Aug 20, 2013 at 20:23

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It seems to me that this is conjectured by Schinzel. His conjecture for this special case is as follows:

Conjecture(Schinzel 1995): For every algebraic number field $K$ there exists a constant $C_1(K)$ such that, if $n > 2$, $A,B ∈ K^{*}$ and the trinomial $x^n +Ax +B$ is reducible over $K$ then either it has a proper factor of degree $≤ 2$ or $n ≤ C_1(K)$. (For $K = \mathbb{Q}$ we have $C_1(\mathbb{Q}) ≥ 52$.)

In other words, we should find an integer $n\ge C_1(K)$, such that $x^n+x+B$ is irreducible. The reference is: A. Schinzel: "Solved and unsolved problems on polynomials", $1995$.

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    $\begingroup$ Of course, Schinzel's conjecture is much stronger than what was asked. Also, in his 1993 paper "On reducible trinomials", he makes this stronger conjecture: for every number field $K$, there exists $C_1(K)$ such that if $n>m$ are positive integers for which $n/\text{gcd}(n,m)>C_1(K)$, then for any $A,B\in K^*$ the polynomial $x^n+Ax^m+B$ is reducible over $K$ if and only if it has a factor of degree $\le 2$. However, it should be noted that this conjecture is motivated by Thm 6 of that paper, and the Math.Review says that the proof of Thm 6 is wrong. $\endgroup$
    – Michael Zieve
    Commented Aug 20, 2013 at 22:01

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