# How to show this ideal is not principal

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.

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There are exceptions when genus theory is strong enough to tell you something; this requires completely ramified prime ideals. Assume e.g. that $(p) = P^3$ and write $(2,s+1) = (\alpha)$. Then $\alpha \equiv a \bmod P$ for some integer $a$, hence $a^3 \equiv \pm 2 \bmod p$ in the integers. If you're lucky, $2$ is not a cube modulo $p$, and then the ideal is not principal.
But how does it follow that $a^3 \equiv \pm2$ mod $p$? We could have that $\alpha^3 = 2u$, with $u$ a unit right? Thanks! – Willem Beek Dec 11 '15 at 20:27