Perfect fields: every irreducible polynomial is separable

Depends on

• Repeated roots in extension fields and separable polynomials
• Field

Used by

• Every algebraic extension of a perfect field is separable Corollary
• The abstract residue computes the coefficient-trace residue at every closed point Corollary
• Delta invariant of a curve singularity Definition
• Geometric genus of a singular curve Definition
• Residue of a rational differential at a separable closed point Definition
• One cocycle carried through the residue realization of Serre duality Example
• Residues on the projective line and the vanishing of their sum Example
• Riemann-Hurwitz for a tame double cover with 2r branch points Example
• x⁵-6x+3 over ℚ is not solvable by radicals Example
• A uniformizer differential generates the module of differentials Lemma
• Free differentials imply regularity in characteristic zero Lemma
• Reduced preparation has nonzero discriminant Lemma
• Separable generation after finite purely inseparable extensions Lemma
• A field is perfect exactly when it has characteristic zero or its Frobenius map is surjective Theorem
• A normal curve over a perfect field is nonsingular Theorem
• An algebraic extension is purely inseparable over its separable closure Theorem
• Dense regular loci on every component Theorem
• Jacobian criterion and openness of the regular locus over a perfect field Theorem
• Openness of the regular locus over a perfect field Theorem
• Over a perfect field, every endomorphism has a unique commuting semisimple-plus-nilpotent decomposition, polynomial in the endomorphism Theorem
• Pure inseparability and its conjugate, embedding, and separable-degree criteria Theorem
• Regular algebras over a perfect field are geometrically regular Theorem
• Smooth proper curves, dominant morphisms and function fields Theorem
• Smoothness over a characteristic-zero field via free differentials Theorem

Dependency tree · two levels

6 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)