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

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

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

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

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

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

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

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

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.