RefacteThe mechanism
Choose a module basis that exposes composition
A divisibility condition in two variables can conceal a one-variable composition. The proof expands in a suitable free-module basis and compares the surviving coefficients along the diagonal. The new working coordinates isolate the restriction imposed by divisibility.
What becomes elementary: A polynomial divisibility question becomes a controlled remainder and coefficient calculation.
Read the free-module proof first. The Galois and Frobenius companion supplies a second route with its own hypotheses; compare which work each representation makes immediate.
For another construction, explore the mechanism calculi. Edition dates, sources and reference chapters identify the scope of each study.
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.