The argument uses a residue expansion in a free module, followed by a reduction along the diagonal. A companion exposition revisits the same criterion through Galois theory and Frobenius descent.
Background
Polynomial rings and basic field theory.
Read the full exposition
The complete argument is available as a PDF.
Revision history ·
Revised September 22, 2026: added a full fiber-constancy criterion over arbitrary fields, using Hasse derivatives to retain inseparable multiplicities. The original composition theorem and its free-module proof are unchanged. The PDF and TeX source agree.