The ramification index measures multiplicity in ramified maps and field extensions. For a finite
extension 
of local fields, it is the positive integer 
defined by
 |
where 
and 
are the maximal ideals in the corresponding valuation
rings.
For a point 
with 
,
the ramification index of 
at 
is a positive integer 
such that there is some open
neighborhood 
of 
so that 
has only one preimage in 
, i.e., 
, and for all other points 
, 
. In other words, the map from 
to 
is 
to 1 except at 
. At all but finitely many points of 
, 
. For any point 
, 
. The ramification index of 
at 
is sometimes called the valency of 
.
