TITLE: Irreducibility of a cubic in $\mathbb{Q}[X]$
QUESTION [5 upvotes]: Problem
Show that $$X^3-2008X^2+2010X-2009$$ is irreducible in $\mathbb{Q}[X]$.

Progress
I considered applying Eisenstein's Theorem, but there are no primes $p$ such that $p|2008$, $p|2009$ and $p|2010$. (This is quite clear as $\nexists p$ prime such that p divides consecutive integers for any choice of p.)
I think this may require an application of Gauss's Lemma, but I'm yet to successfully show it. Any help would be appreciated.
Regards.

REPLY [11 votes]: HINT $\ $ By Gauss's lemma if $\rm\:f(x)\:$ splits over $\rm\:\mathbb Q\:$ then it splits over $\rm\:\mathbb Z\:,\:$ hence it splits over $\rm\:\mathbb Z/3\:,\:$ contra $\rm\ f(x)\ \equiv\ x^3-x^2+1\ $ has no roots $\rm\ mod\ 3\:.$
REMARK $\ $ Gauss's Lemma is a bit overkill here. But it's trivial to prove the simple monic case:
LEMMA $\rm\ \ f\ g\in \mathbb Z[x]\ \Rightarrow\ g \in \mathbb Z[x]\ \ \ if\ \ \ g\in \mathbb Q[x],\ \ monic\ f\in \mathbb Z[x]$
Proof $\ $ Write $\rm\ g = c\ x^n + g'\:,\  \ deg\ g' < deg\ g = n\:.\ $ Since $\rm\:f\:g \in \mathbb Z[x]\:$ its lead coeff $\rm\:c\in\mathbb Z\:.\:$ Thus $\rm\:f\ g' =\: f\ g - c\ x^n\:f\: \in\: \mathbb Z[x]\:,\:$ so $\rm\:g'\in \mathbb Z[x]\:$ by induction on deg $\rm\:g\:,\:$ so $\rm\ g = c\ x^n + g'\in \mathbb Z[x]\:.\:  $ QED