Eisenstein Criterion with Polynomials of Rational Coefficients? I'm wondering if it's possible to apply the Eisenstein criterion to polynomials that have rational, non-integer coefficients? I am thinking about using a shift to go to integers and apply the Eisenstein criterion there. Then, since f(x) is irreducible iff f(x+c) is irreducible, that should show that the original polynomial is irreducible. For instance, with the polynomial $x^5-(3/5)x^4+9x-3$, I wonder if there's some way to shift this so it gets integer coefficients and we can do the procedure I said?
 A: That polynomial is Eisenstein at $3$.  It is actually irrelevant what other prime factors are doing in the coefficients, whether they are in numerators or denominators.  A way to deal with this is to give up on trying to work in $\mathbf Z[x]$ and instead allow yourself to invert all primes other than $3$: let the coefficients be in the ring $\mathbf Z_{(3)}$, which denotes fractions $a/b$ where $(3,b) = 1$. In this ring (a "localization" at the prime 3), which has fraction field $\mathbf Q$, the only prime element is $3$ and the Eisenstein criterion applies with the prime $3$ (not with other primes!).
Think about it this way: if $A$ is a UFD with a prime element $p$, the notion of Eisenstein polynomials at $p$ in $A[x]$ makes sense and such polynomials are irreducible in $A[x]$ and in $K[x]$ where $K$ is the fraction field of $A$.  Can you prove that? In your example, take $A$ to be not $\mathbf Z$, but $\mathbf Z_{(3)}$ (a UFD with just one prime element up to unit multiple, namely $3$) and $K = \mathbf Q$.
A: Just multiply  the polynomial by the common denominator of the coefficients. Eisenstein criterion works not only for monic polynomials, but also polynomials with leading coefficient not divisible by $p$.
