This is related to Chpt 4, Sec 4 of Taylor Frohlich, Algebraic Number Theory.
From Ostrowski theorem, any absolute value over $\mathbb Q$ are either equivalent to some $p$-adic absolute value or standard absolute value.
$S$ is a finite set of inequivalent of absolute values of $K$ which includes only all Archimedean absolute values.
In particular, I pick $S$ coinciding with the set of all Archimedean absolute values extending $\mathbb Q$'s Archimedean absolute values. Then $U_K(S)=\{a\in K^\star \mid \forall v\not\in S,\ |a|_v=1\}=\mu_K\times \mathbb Z^{|S|-1}$. Here $K$ is a number field.
$\textbf{Q:}$ For $K / \mathbb Q$, it is clear that there are only finitely many absolute values extending the Archimedean absolute value $|x|$. In general, is there an example with infinite number Archimedean absolute values for a finite extension over some field other than $\mathbb Q$? (It is clear it cannot be extension of number fields or between number fields.)
$\textbf{Q':}$ My guess is that $\mathbb C/ \mathbb Q$ has infinite Archimedean absolute values extending over $\mathbb Q$. However, I am not sure this is correct in the sense that $\mathbb C$ should be realized as inverse limit of finite Galois group associated to $\mathbb C$ as a field.