Why is this result true :
$K$ being a local field with algebraicaly closed residue field then an abelian extension of $K$ (let's say $L$) is totaly ramified!
I didn't find any reference in all the books or course i read.
It's exercise 7.6 p 186 in N. Childress "Class Field Theory" This exrcise is juste after the "Decomposition theorem" If $K$ is a local field and $L/K$ is finite galois extension then there is a totaly ramified subextension $L'$ with $L'^{ur}=L^{ur}=LK^{ur}=L'K^{ur}$ ($ur$ for "maximal unramified").
I have no idea especially after a result which assume explicitly the resifue field to be finite.