The irreducibility of polynomials for specific cases I'm trying to prove the following are irreducible over $Q$.
$f = 7x^4-18x^3+6x^2-24x+12$
and 
$g = 2x^3-5x+25$
For the first I have said that f is primitive and used Eisenstein's criterion with p=3. Then, by Gauss' lemma, f is irreducible over $Q[x]$. Is this sufficient for irreducibility over $Q$?
Further, for g I cannot find a suitable value of p. How can I resolve this?
 A: Eisenstein gives you conditions for irreducibility, so I'm not sure why you mention the primitive thing.
For the second one: perhaps the easiest way (at least for me and in this particular case) is to use the rational root test: if $\;\frac ab\;$ is a rational root of $\;g(x)\;$ , then $\;a\mid25\;,\;\;b\mid 2\;$, and since a polynomial of degree $\;\le 3\;$ over some field is reducible iff it has a root in that field, we get the options are
$$\pm\left\{\;1\,,5,\,25,\,\frac12,\,\frac52,\,\frac{25}2\;\right\}$$
and now it is easy, though maybe a little boring, to check none of this is a root.
A: I'm not really what your asking in your first question.  When someone says a polynomial $f$ is irreducible over a field $F$ that generally means that $f$ is an irreducible element in $F[x]$.  This is given directly by Eisensteins criterion though, not sure what you need Gauss for.
For your second question, there indeed is not a prime for which eisensteins criterion is going to work on $g$.  But the polynomial is degree $3$, so if it's reducible then at least one factor is degree $1$, meaning $g$ has a root.  So just check that $g$ has no roots.
