Suppose we have a quadratic (Galois) extension of $\mathbb{Q}$, call it $k$ with Galois group $G$. If we look at the ring of integers inside of $k$, call it $\mathcal{O}_k$, is it true that $\mathcal{O}_k$ is stable under $G$? That is, do we have that $\sigma x\in\mathcal{O}_k$ for every $x\in \mathcal{O}_k$ and $\sigma\in G$?
I don't even really know what the routes are that someone would take to look at such a question. I guess this is probably true since elements of the ring of integers are solutions to certain (monic) polynomials with integer coefficients, so the galois group would just change to a different solution of the polynomial, hence remaining in $\mathcal{O}_k$?