Order not preserved by Galois Let $K/\mathbb Q$ be a Galois extension. I'm trying to come up with an example of an order (necessarily nonmaximal) in $K$ which is not preserved by Galois. For simplicity, I've looked at monogenic orders $\mathcal O_K [\alpha]$ where the question boils down to finding a case where $\mathbb Z[\alpha]$ does not contain all the conjugates of $\alpha$.
Working locally, it seems like if the valuation of $\alpha$ is very large, it should be unlikely that a given conjugate $\alpha'$ of $\alpha$ will be in $\mathbb Z_p[\alpha]$. My thinking is that here, $\alpha$ and $\alpha'$ differ by a unit, but if $v(\alpha)$ is large then the order is a lot smaller, and becomes less and less likely to contain the necessary unit. However, I haven't been able to work out an example along these lines.
Edit: the current answer is good, but I would be more interested in one in the local case that I outlined.

$$O=\Bbb{Z}+(1+2\zeta_8+2i)\Bbb{Z}[\zeta_8]\qquad \ne \qquad O'=\Bbb{Z}+(1+2\zeta_8^{-1}-2i)\Bbb{Z}[\zeta_8]$$ $(1+2\zeta_8+2i)$ is a prime ideal above $3$ and residue field $\Bbb{F}_9$.
The image of $O$ in $\Bbb{Z}[\zeta_8]/(3,1+2\zeta_8+2i)$ has 3 elements while the image of its complex conjugate order $O'$ has 9 elements.
Something similar will work whenever $K$ is not a quadratic field.
