##  [Purely Inseparable Extension](/purely-inseparable-extension-0) 

 Definition

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

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

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

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

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

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

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

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

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.