The irreducibility of polynomials for specific cases

Question

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?

Answer 1

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.

Answer 2

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.