## [Inseparable Extension](/inseparable-extension-0) Definition 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 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 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 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 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 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 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 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 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.