This question was inspired by [this one](https://mathoverflow.net/questions/56506/irreducibility-of-some-trinomials-modulo-p). 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?



About irreducible trinomials
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


