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.
Irreducible polynomial $\frac{x^{n}+x^{m}-2}{x^{\gcd(n,m)}-1}$ over $\mathbb{Q}$.
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abstract-algebra
field-theory
galois-theory
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1Here is a generalization: http://mathproblems123.wordpress.com/2009/11/02/irreductible-polynomial/ And here is a useful lemma: http://mathproblems123.wordpress.com/2009/11/09/position-of-roots/ – 2012-10-24