My Reference is https://virtualmath1.stanford.edu/~conrad/249BW09Page/handouts/cfthistory.pdf In chapter 7, page 17, Keith says
For example, $(\alpha, L/K)_v$ lies in the common decomposition group $D(w|v)$ for all places $w|v$ on $L$
for unramified places $v$ it follows from definition, but it feels that for ramified primes there is something non-trivial to say, or at least I don't see why it is obvious.
Can someone point me in a direction for showing that?