Does this exact sequence split?

Question

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.

Answer 1

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). 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.