##  [Ramification Point](/ramification-point-0) 

 Definition

A point of a finite morphism (or extension) at which the map fails to be étale or locally bijective, measured by a ramification index greater than one; algebraically this corresponds to nontrivial inertia or nontrivial extension of residue fields.

 

 

 

 

 

 





## Principle

Principle

Ramification points occur where the local algebra of the map acquires nilpotent or repeated factors in the fiber, equivalently where valuation-theoretic ramification index or inertia subgroup is nontrivial, causing deviation from local étaleness.

 

 

 

 

 





## Demonstration

Demonstration

For the map on coordinate rings k[t] → k[u] sending t to u^e, the image of u = 0 is a ramification point of index e: the fiber has multiplicity e, the differential vanishes to order e−1, and the residue field extension is trivial while the local degree increases.

 

 

 

 

## Misapplication

Misapplication

Mistaking a point where the target has a singularity for a ramification point of the morphism is a misuse: ramification concerns the behaviour of the source over the base, not intrinsic singularities of the base itself.

 

 

 

 

 





## Consequence

Consequence

Locating ramification points identifies where local invariants like the different, discriminant, and conductor are supported, influences Riemann–Hurwitz-type formulas, and controls how local Galois or inertia groups act.

 

 

 

 

## Reversal

Reversal

An étale point (unramified point) is one where the morphism is flat and unramified: the ramification index is one, the differential is nonvanishing, and local rings map as separable extensions.

 

 

 

 

 





## Boundary

Boundary

Applicable to finite morphisms of schemes, finite extensions of local fields, and algebraic coverings; excludes topological branch phenomena that do not carry algebraic inertia data and must account for inseparability in positive characteristic.

 

 

 

 

 





## Semantic Tension

Semantic Tension

The algebraic term 'ramification point' overlaps with the analytic 'branch point' but emphasizes inertia, indices and discriminants rather than monodromy; in arithmetic contexts it also splits into tame versus wild behaviors.

 

 

 

 

 





## Synthesis

Synthesis

A ramification point is a base-point of a finite algebraic map where étaleness fails: local degrees jump, inertia acts nontrivially, and algebraic invariants such as the different and discriminant record the deviation from being unramified.