TITLE: Is the ring of integers in a relative algebraic number field faithfully flat over a ground ring?
QUESTION [7 upvotes]: Let $L$ be a finite extension of an algebraic number field $K$.
Let $A$ and $B$ be the rings of integers in $K$ and $L$ respectively.
Is $B$ faithfully flat over $A$?
What if $L$ is an infinite algebraic extension of $K$?

REPLY [5 votes]: A module over a Dedekind domain is flat if and only if it is torsion-free. So any extension of rings $A\rightarrow B$ with $A$ Dedekind and $B$ a domain is flat. In particular, if the extension is integral, it is faithfully flat (by lying-over, as you point out in your comment).