Let $K$ be a number field. $O\_K$ be its ring of integers, so $O\_K^\*$ are the units.
We have sequence


$1 \rightarrow O\_K^\* \rightarrow K^\* \rightarrow K^\*/O\_K^\* \rightarrow 1$


Note that $K^\*/O\_K^\*$ is essentially the group of principal fractional ideals.


Does this sequence split for all number fields? It seems true for $K = Q$ and
$Q[i]$. But I doubt that it is always true for every number field.



Does this exact sequence split?
The exact sequence in the original post splits for every number field $K$. To see this, let $P\_K$ be the multiplicative group of nonzero principal fractional ideals of $K$, and let $I\_K$ be the multiplicative group of all nonzero fractional ideals of $K$. Clearly, $P\_K$ is a subgroup of $I\_K$, and it is isomorphic to $K^\*/O\_K^\*$. As $I\_K$ is free abelian (the nonzero prime ideals of $O\_K$ form a free generating set), $P\_K$ is also free abelian (see [here](https://en.wikipedia.org/w/index.php?title=Free_abelian_group&oldid=317358888#Subgroup_closure)). Let $X$ be a set of free generators of $P\_K$, and write each $x\in X$ as $(f(x))$ with a suitable $f(x)\in K^\*$. The map $f:X\to K^\*$ extends uniquely to a homomorphism $P\_K\to K^\*$, and it splits the exact sequence in the original post.


