$H^{1}(\mathbb{Q},\mathbb{Z})=?$

Question

I have the following questions:

(1.) $H^{1}(\mathbb{Q},\mathbb{Z})=?$

(2.) If $k\subseteq\overline{\mathbb{Q}}$ is a field, what is $H^{1}(k,\mathbb{Z})$?

(3.) If $R$ is an integral domain with field of fractions $k(R)$, what is $H^{1}(k(R),R)$?

I am particularly interested in cases (1.) and (2.). I want to know whether or not every short exact sequence ending in $\mathbb{Q}$ or $k$ splits.

Answer 1

Hints (for (1) and (2)):

• Let $k$ be any field, acting on $\mathbb{Z}$ trivially. Then $H^1(k,\mathbb{Z})=Hom_{grp}(k,\mathbb{Z})$.

• If $f:k\to\mathbb{Z}$ is a group morphism, show that for any prime $p\neq char(k)$, $Im(f)\subset p\mathbb{Z}$.

  • Conclude.