Let $K$ be an extension of $\mathbb Q$. $\mathcal O_K$ is the set of all the elements of $K$ which are integral over $\mathbb Z$.
Now suppose $[K:\mathbb Q]=m$ and $\{e_i\}_{i=1}^m$ is an integral basis of $\mathcal O_K$. Let $\mu\in Gal(K/\mathbb Q)$ and $\{\mu (e_i)\}_{i=1}^n$ are the images of $\{e_i\}_{i=1}^m$ under the action of $\mu$.
Now I am in doubt whether $\{\mu (e_i)\}_{i=1}^n$ are also an integral basis of $\mathcal O_K$?
Anyone know the answer?
Thank you for your help.