A field extension L/K in characteristic p>0 such that every element of L is algebraic over K and for each alpha in L there exists n≥0 with alpha^{p^n} in K; equivalently the separable subextension is trivial and all minimal polynomials are pure p-power polynomials.

Principle

Occurs from the failure of the derivative test in positive characteristic: Frobenius powers collapse distinct roots, so algebraicity can come solely from p-th power relations rather than separable minimal polynomials.

Demonstration

Let K = F_p(t^p) inside L = F_p(t). Here t^p ∈ K but t ∉ K, and every element of L satisfies a pth-power relation over K (e.g. t^{p} ∈ K), so L/K is purely inseparable.

Misapplication

Calling an extension purely inseparable when it merely contains inseparable elements mixed with nontrivial separable subextensions; confusing 'inseparable' with 'not separable' generically can mask an underlying separable part.

Consequence

Galois-theoretic tools fail: there is no nontrivial separable Galois group, field embeddings over K are not distinct, and geometric maps induced by such extensions are inseparable morphisms with ramifications for dimension and differential forms.

Reversal

A separable extension, where minimal polynomials have distinct roots and Frobenius does not collapse embeddings; separable and purely inseparable are complementary parts of the algebraic closure decomposition.

Boundary

Only meaningful for algebraic extensions in characteristic p>0; does not apply to characteristic zero, transcendental extensions, or to extensions that decompose nontrivially into separable and inseparable components.

Semantic Tension

Between 'inseparable' (some elements may be inseparable) and 'purely inseparable' (every element satisfies p^n-power relation over base); nearby are notions of separable closure and perfect fields.

Synthesis

A purely inseparable extension is an algebraic extension in positive characteristic generated entirely by p-power relations: Frobenius-induced collapse produces no separable structure, leading to specific failures of classical Galois and differential behavior.