Q:

Is an integral basis $\{e_i\}$ of $\mathcal O_K$ under the action of $\mu\in Gal(K/\mathbb Q)$ also an integral basis?

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.

A:

The subring $\mathcal O_K$ is stable under $Gal(K/\mathbb Q)$, as one easily verifies.  Thus $\mu$ is an automorphism of $\mathcal O_K$ (as a ring, and in particular as a $\mathbb Z$-module).  The answer to your question is now seen to be yes, since any automorphism of a free $\mathbb Z$-module will take one basis to another basis.

