I have been brushing up on cubic number fields. Specifically, let $s$ be a root of the polynomial $x^3 + x^2 + 3x + 17$, and consider $K = \mathbb{Q}(s)$; we have $\mathcal{O}_K = \mathbb{Z}[s]$, and
$2\mathcal{O}_K = (2,s+1)^3,$
this by the theorem of Dedekind. My favourite computer algebra system tells me that $(2,s+1)$ is not principal, but I would like to justify this myself.
How may I show that this ideal is not principal?
I can see how my question is equivalent to asking why the ring $\mathbb{Z}[s]$ has no element of norm $\pm 2$, or why 2 is irreducible in this ring. In principle I could answer this by writing down the norm of a general element $a + bs + cs^2$ for $a,b,c \in \mathbb{Z}$; but this is something that I really do not want to do. Rather I'm looking for a way to answer the question whilst keeping my hands clean.