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This question was inspired by this one. For every $n>m>0$ consider the polynomial $p_{m,n}=x^n-x^m-1$.

For which $m,n$ is $p_{m,n}$ irreducible over $\mathbb Q$?

In particular, if $m$ is odd, is it always irreducible?

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by axccident I found your pos and the following article: on the irrecucibility of the trinomias $x^n \pm x^m \pm 1$, Helge Trevberg, Math. Scand 8 (1960) 121-126 –  miracle173 Nov 8 '13 at 2:16

1 Answer 1

up vote 20 down vote accepted

MR0124313 (23 #A1627) Ljunggren, Wilhelm On the irreducibility of certain trinomials and quadrinomials. Math. Scand. 8 1960 65–70. 12.30

The author considers the irreducibility over the field of rational numbers of the polynomials $f(x)=x^n+ε_1x^m+ε_2x^p+ε_3$, where $ε_1,ε_2,ε_3$ take the values $\pm1$. He proves that if $f(x)$ has no zeros which are roots of unity, then $f(x)$ is irreducible; if $f(x)$ has exactly $q$ such zeros, then $f(x)$ can be factored into two factors with rational coefficients, one of which is of degree $q$ with all these roots of unity as zeros, while the other is irreducible (and possibly merely a constant). He also determines all possible cases where roots of unity can be zeros of $f(x)$. As a corollary he is able to give a complete treatment of the trinomial $g(x)=x^n+ε_1x^m+ε_2$, where $ε_1,ε_2$ take the values $\pm1$. The irreducibility of this trinomial was studied by E. S. Selmer, who gave a partial solution [Math. Scand. 4 (1956), 287--302; MR0085223 (19,7f); see also #A1628]. The methods used are direct and elementary. Reviewed by H. W. Brinkmann

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@Gerry: Thanks! That was quick! –  Mark Sapir Feb 25 '11 at 0:13

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