Ramification in a finite extension or morphism where the ramification index is prime to the residue characteristic and higher ramification groups vanish; inertia is of order prime to p and local behaviour is controlled and well behaved.

Principle

Tame ramification is characterized by inertia groups of order prime to the residue characteristic and trivial higher ramification filtrations, which makes many local invariants multiplicative and enables classical formulas and descent arguments.

Demonstration

A Kummer extension of local fields obtained by adjoining an e-th root with e prime to the residue characteristic is tamely ramified: the inertia is cyclic of order e, higher ramification groups vanish, and the conductor is minimal.

Misapplication

Calling a ramification tame merely because it has small numerical index is wrong; one must check that the index is prime to the residue characteristic and that higher ramification subgroups are trivial.

Consequence

Tame ramification simplifies the arithmetic: discriminants and conductors satisfy simpler relations, Galois module structure is more tractable, and deformation and cohomological calculations are generally easier than in the wild case.

Reversal

Wild ramification is the inverse case: p‑power inertia, nontrivial higher ramification groups, and fractional jumps complicate the arithmetic and break many tame simplifications.

Boundary

Applies when residue characteristic does not divide ramification indices and separability conditions hold; excludes inseparable or p‑power inertia phenomena and topological branch points lacking arithmetic residue data.

Semantic Tension

The adjective 'tame' suggests simplicity, but mathematically it is a precise condition (prime-to-p inertia and vanishing higher groups); confusion arises when practitioners conflate numerical smallness with tameness.

Synthesis

Tame ramification denotes the class of ramification with prime‑to‑p inertia and trivial higher ramification subgroups, giving controlled local behaviour and permitting the use of classical invariants and formulas absent in wild situations.