Showing $x^{5}-ax-1\in\mathbb{Z}[x]$ is irreducible

Solution 1:

Let $P=X^5-aX-1$. If $P$ had a rational root $\frac{p}{q}$ with $p$ and $q$ coprime, we would deduce $p^5-apq^4-q^5=0$. Then $p$ divides $p^5-apq^4$, we see that $p$ divides $q^5$. By iterating Gauss' lemma, $p$ divides $q^4,q^3,q^2$ etc and finally $p$ divides $1$, so $p=\pm 1$. Similarly $q$ divides $p^5$ so $q=\pm 1$. So the only rational roots possible are $-1$ (corresponding to $a=2$) and $1$ (corresponding to $a=0$). So $P$ has no degree $1$ factors.

So all we've left to show is that there is no factor of degree $2$. So assume $P=UV$, with $U,V \in {\mathbb Z}[X]$ and ${\sf deg}(U)=3, {\sf deg}(V)=2$. Since $P$ is monic, $U$ and $V$ are monic also (we replace them by their opposites if necessary).

Write $U=X^3+u_2X^2+u_1X+u_0$ and $V=X^2+v_1X+v_0$. Then,

$$ P=UV=X^5+(u_2+v_1)X^4+(u_2v_1+u_1+v_0)X^3+(u_2v_0+u_1v_1+u_0)X^2+(u_1v_0+u_0v_1)X+u_0v_0 $$ Identifying the coefficients in $X^4,X^3,X^2$, we express $u_0,u_1,u_2$ in terms of the other coefficients : $$ u_2=-v_1,u_1=v_1^2-v_0,u_0=2v_0v_1-v_1^3 $$ Then the product $UV$ becomes $$ UV=X^5-(v_1^4-3v_0v_1^2+v_0^2)X+(2v_0^2v_1-v_0v_1^3) $$ The constant coefficient can be factorized as $v_0v_1(2v_0-v_1^2)$. So $v_0,v_1$ and $2v_0-v_1^2$ must all be equal to $1$ or $-1$. So necessarily $v_0=1,v_1=-1$, and hence $U=X^3+X^2-1,V=X^2-X+1,a=-1$.