Let $k$ be a number field with $[k:\mathbb Q]=n=r_1+2r_2$, where $r_1$ is the number of the real embeddings, and $r_2$ the number of pairs of complex embeddings. So we have $r_1+r_2$ number of Archimedean valuations $\{v:v\mid\infty\}$.
Let $k_\infty=\prod_{v\mid\infty} k_v$. For $t\in\mathbb R^\times_{+}$, we have $\prod_{v\mid\infty}t\in\prod_{v\mid\infty}k_v$.
My problem is the following. what is the value of $\prod_{v\mid \infty}|t|_v?$ Is it equal to $t^{n}$?