Separable algebraic elements and separable extensions

Depends on

• Repeated roots in extension fields and separable polynomials
• The evaluation kernel and the unique monic irreducible minimal polynomial of an algebraic element
• Algebraic and transcendental elements and algebraic extensions

Used by

• A finite Galois extension is Galois over every intermediate field Corollary
• An extension that is both separable and purely inseparable is trivial Corollary
• Every algebraic extension of a perfect field is separable Corollary
• Finite Galois extensions and Gal(K/F) Definition
• Separating transcendence basis and separably generated extensions Definition
• Finite field extensions and etaleness Example
• Finite power map of the affine line Example
• Finite separable extensions have zero Omega Example
• A dense hypersurface chart with a nonzero partial derivative Lemma
• A finite normal extension is separable over its purely inseparable fixed field Lemma
• Finite-type field extensions with zero Ω Lemma
• Separable generation after finite purely inseparable extensions Lemma
• Separable residue and the cotangent sequence of a local algebra Lemma
• Unramified residue extensions are finite separable Lemma
• A finite extension generated by elements all but possibly one of which are separable is simple Theorem
• A finite extension is separable if and only if [K:F]ₛ=[K:F] Theorem
• An algebraic extension generated by separable elements is separable Theorem
• Finite separable extensions have finite minimal Galois closures Theorem
• Finitely generated extensions of a perfect field are separably generated Theorem
• Norm and trace from embeddings, with the inseparable exponent in the norm formula Theorem
• Polynomial algebras over fields have finite integral closures Theorem
• Separability is transitive in towers of algebraic extensions Theorem
• The trace form of a finite extension is nondegenerate exactly when the extension is separable Theorem

Dependency tree · two levels

9 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• P. L. Clark, Field Theory, Chapters 3 to 5 (standard reference, not scraped)
• J. S. Milne, Fields and Galois Theory, Chapters 2, 3, and 5 (standard reference, not scraped)