Alphabeta MathHow statement and proof provenance work
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Fields of characteristic zero, finite fields, and algebraically closed fields are perfect
Statement
Every field of characteristic zero is perfect. Every finite field is perfect, and every algebraically closed field is perfect.
Facts & Assumptions
Given: A field in one of the classes named in the Statement.
Perfectness is equivalent to characteristic zero or, in characteristic , surjectivity of Frobenius (A field is perfect exactly when it has characteristic zero or its Frobenius map is surjective).
Frobenius is an automorphism of every finite field (Frobenius is an injective endomorphism in characteristic , and an automorphism for finite fields).
In an algebraically closed field, every nonconstant polynomial has a root (An algebraically closed field: every nonconstant polynomial has a root in the field).
Proof
The characteristic-zero case is immediate from [L1].
If is finite of characteristic , [L2] makes Frobenius surjective, so [L1] makes perfect.
If is algebraically closed of characteristic , then for every the polynomial has a root by [L3]; hence every is a th power and [L1] makes perfect.
Depends on
Used by
- General hypersurfaces give smooth complete intersections Corollary
- Generic smoothness on the source Corollary
- Minimal tangent dimension and homogeneous regularity Corollary
- ℚ(³√2)/ℚ is separable and nonnormal with trivial automorphism group Counterexample
- A nondegenerate projective quadric Example
- ℚ(³√2) has three embeddings into ℚ̄ but only one ℚ-automorphism Example
- x⁵-6x+3 over ℚ is not solvable by radicals Example
- A dense hypersurface chart with a nonzero partial derivative Lemma
- A dominant map has a surjective differential on a dense source open Lemma
- A tangent direction is realized by a local smooth curve Lemma
- Finite-type field extensions with zero Ω Lemma
- Flat maps with geometrically regular fibres have standard smooth local presentations Lemma
- Separable generation after finite purely inseparable extensions Lemma
- An algebraic extension containing a root of every nonconstant base polynomial is algebraically closed Theorem
- Bertini smoothness away from the base locus Theorem
- Generic smoothness over a dense target open Theorem
- The complex numbers are algebraically closed Theorem
Dependency tree · two levels
15 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)