Let $k$ be a field with absolute value $|\cdot |$ and let $S^1(k) := \lbrace x \in k \mid |x|=1\rbrace \le k^\times$. Since the absolute value definies a metric on $k$, $S^1(k)$ is a topological group. In particular, $S^1(k)$ has a classifying space $BS^1(k)$.

**Question 1:** What is known about $BS^1(k)$ ?

In case $k=\mathbb R$ (with the usual absolute value), $S^1(\mathbb R)=\lbrace \pm 1 \rbrace \cong \mathbb{Z}/2$ and $BS^1(\mathbb R) = P^\infty(\mathbb R)$.

Similarly, if $k=\mathbb C$ (with the usual absolute value), $S^1(\mathbb C)=S^1$ and $BS^1(\mathbb C) = P^\infty(\mathbb C)$. This motivates:

**Question 2:** Is it known under which conditions the relation $BS^1(k) = P^\infty(k)$ holds ?