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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.

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Yes. Hint: think about the discriminant of an $n$-tuple of elements of $\mathcal{O}_K$. – Pete L. Clark Dec 16 '11 at 16:03
up vote 8 down vote accepted

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.

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