An algebraic field extension K ⊂ L in which at least one element of L has a minimal polynomial over K with repeated roots (equivalently, the minimal polynomial and its formal derivative are not coprime), so the extension fails the separability condition.

Principle

Separable behaviour requires distinct roots of minimal polynomials; inseparability arises in positive characteristic when formal derivatives vanish and distinct-root criteria collapse, producing purely inseparable or mixed algebraic extensions.

Demonstration

Over a field K of characteristic p > 0, the extension K(t^{1/p}) obtained by adjoining a pth root of an element t whose pth root is not already in K is inseparable: the minimal polynomial X^p - t has derivative pX^{p-1}=0, so its root is repeated in any splitting field.

Misapplication

Treating every algebraic extension in characteristic p as separable or assuming Galois correspondence holds unchanged; or using separability-based arguments (e.g. counting automorphisms equal to degree) without checking for inseparable elements.

Consequence

When correctly identified, inseparability alters the structure of field automorphism groups, invalidates naive Galois correspondences, and forces the use of purely inseparable descent and Frobenius-based techniques (for example, reducing arguments to separable subextensions or passing to perfect closures).

Reversal

A separable extension is the inverse concept: every element's minimal polynomial has distinct roots and separability restores usual Galois theory statements such as degree equaling number of K-embeddings into an algebraic closure.

Boundary

Applies only to algebraic extensions of fields; transcendental extensions are not classified as separable/inseparable in this sense. The phenomenon is relevant chiefly in positive characteristic; in characteristic zero every algebraic extension is separable.

Semantic Tension

Closely related but distinct from the notion of a polynomial being inseparable: a polynomial can be inseparable over a field if its derivative vanishes, while an extension is inseparable if it contains at least one such element — the two are related but address polynomial versus extension-level properties.

Synthesis

An inseparable extension is an algebraic field extension, occurring in positive characteristic, characterized by minimal polynomials with repeated roots; recognizing it redirects methods from classical separable/Galois techniques toward Frobenius and purely inseparable constructions.