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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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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