Wild ramification is ramification in which the residue characteristic divides the ramification
index. More precisely, let 
be a finite Galois
extension of local fields, with residue characteristic
. If 
is the Galois group and 
is its inertia
subgroup, the wild inertia subgroup 
is the first higher ramification
group 
.
It is the unique Sylow p-subgroup of 
, and the quotient
group 
is a cyclic group of order prime to 
. Thus there is an exact sequence
 |
The field extension 
is tamely ramified
when 
is trivial and wildly ramified when 
is nontrivial.
The higher ramification groups measure how strongly elements of 
approach the identity. In lower numbering, they are defined for 
by
 |
Here 
is the normalized discrete valuation on 
, and 
is its valuation ring.
Their contribution to the Artin conductor of a
Galois representation is measured by the
Swan conductor. For an elliptic
curve, this wild contribution to the local elliptic
curve conductor vanishes at residue characteristics greater than 3, but it can
be nonzero at 2 and 3. This accounts for the extra local
conductor exponents found by Tate's algorithm
at those primes.
