A polynomial over a field of positive characteristic whose formal derivative is identically zero (or, more generally, whose roots in an algebraic closure are inseparable), so that roots occur with inseparability and the polynomial induces inseparable field extensions.

Principle

In positive characteristic p, the vanishing of the formal derivative indicates that the polynomial is a polynomial in x^{p} (after change of variables) or is purely inseparable; inseparability obstructs the usual Galois theory and produces extensions without distinct embeddings.

Demonstration

Over a field of characteristic p, the polynomial f(x) = x^{p} - a has derivative 0 and is inseparable when a is not a pth power; more generally x^{p^e} - b produces purely inseparable extensions of exponent e.

Misapplication

Assuming inseparable polynomials behave like separable ones — for instance, counting distinct roots or applying separable Galois descent arguments; inseparable factors require different handling in extension theory and in constructing splitting fields.

Consequence

Inseparable polynomials produce extensions that are not generated by distinct embeddings and force consideration of purely inseparable closure; geometrically they lead to nonreduced fibres and affect étaleness and smoothness criteria.

Reversal

A separable polynomial has nonzero formal derivative and distinct roots in a separable closure; separability restores classical Galois correspondence, simple ramification behaviour, and well‑behaved splitting fields.

Boundary

Inseparability is a phenomenon only in positive characteristic; over characteristic zero every polynomial is separable. The term 'purely inseparable' emphasises that every root has p‑power multiplicity and that the extension degree is a power of p.

Semantic Tension

Tension arises between 'inseparable' as defined by vanishing derivative and 'inseparable' as root multiplicity in a given factorisation; also between 'inseparable' and 'inseparable in families' (behaviour under base change), which requires care.

Synthesis

An inseparable polynomial in characteristic p is one whose derivative vanishes and whose roots cannot be separated by distinct embeddings; it generates purely inseparable extensions that alter arithmetic, Galois, and geometric properties compared with the separable case.