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I have to show that the polynomial $$f(x)=\frac{x^{n}+x^{m}-2}{x^{\gcd(n,m)}-1}$$ is irreducible over $\mathbb{Q}$, for all $n,m \in \mathbb{N}$. Any idea as to how I can show this.

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Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. If this is homework, please add the homework tag; people will still help, so don't worry. – Julian Kuelshammer Oct 24 '12 at 5:54
Here is a generalization:… And here is a useful lemma: – Beni Bogosel Oct 24 '12 at 7:00
@JulianKuelshammer: Since this is an old Miklos Schweitzer problem, I don't think it was given as homework. Although, the mention of the OP's work would make the question more valuable. – Beni Bogosel Oct 24 '12 at 10:43

Hint: Say that $gcd(n,m)=d$, and write $f(x) = \dfrac{x^n+x^m-2}{x^d-1}$ as

$$f(x) = x^{(c-1)d}+x^{(c-2)d} + \cdots x^{bd}+2x^{(b-1)d)}+ \cdots +2x^d +2$$ where $n = cd$, $m=bd$. Consider the polynomial $g(x) = x^{c-1} + \cdots + x^b +2x^{b-1}+ \cdots +2x + 2 $. Show that this polynomial only has roots of absolute value greater than 1. Use this to show that the roots of f satisfies a similar property, and derive a contradiction.

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