Prove that $(5x^3+9x^2-27x+3)$ is a maximal ideal in $\mathbb{Q}[x]$

Question

We have shown that $Q[x]$ is a Euclidean Domain, and thus is a Principal ideal domain. A principal ideal $(f)$ is maximal $\iff$ $f$ is irreducible in $Q[x]$. But how do I show that \begin{equation*} 5x^3+9x^2-27x+3 \end{equation*} is irreducible in $Q[x]$?

That is for some $z,y\in Q[x]$ where \begin{equation*} 5x^3+9x^2-27x+3 = zy \end{equation*} then either $z$ or $y$ is a unit?

Answer 1

As said before, Eisenstein's criterion says that ( in $\mathbb{Z}[x]$ or $\mathbb{Q} [x]$ )

$(f)$ where $f = a_nx^n+a_{n-1}x^{n-1}+...+a_0$ is irreducible if there exists a prime $p$ such that

1. $p \vert a_i$ for all $i \neq n$

2. $p \nmid a_n$

3. $p^2 \nmid a_0$

So for this example, $p=3$ works, therefore your $(f)$ is irreducible and thus maximal.

Answer 2

Since it is degree $3$, if it were not irreducible, then it could be factored into $g(x)f(x)$. One would need to be a degree $1$ polynomial. That is $(x-r)$ for some $r\in \mathbb{Q}$. You can use the rational root theorem to rule out all the rational roots. Or use Eisenstein's criterion (if you have seen this yet), it is way easier.