Let $L/K$ be an extension of number fields and $v$ be a prime (an equivalence class of valuations) of $K$ and $d_L$ the absolute discriminant of $L$.
I know that a rational prime $p$ in $\mathbb Q$ ramifies in $L$ iff $p\mid d_L$.
Is it true that an archimedean prime $v$ ramifies in $L$ if and only if $v\mid d_L$? How can I prove this?
Also then what is the significance of a product of some collection of algebraic objects taken over such $v$'s, for example, something like $\prod_{v\mid d_L}A_{v,w}$ with $A_{v,w}$ complex numbers where for each $v$ a prime $w$ of $L$ is chosen above $v$?
Many thanks for your help.