Alphabeta MathHow statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Every algebraic extension of a perfect field is separable
Statement
If is algebraic and is perfect, then is separable.
Facts & Assumptions
Given: An algebraic extension with perfect.
Every algebraic element has a monic irreducible minimal polynomial over the base (The evaluation kernel and the unique monic irreducible minimal polynomial of an algebraic element).
Every nonconstant irreducible polynomial over a perfect field is separable (Perfect fields: every irreducible polynomial is separable).
An extension is separable when every one of its elements has separable minimal polynomial over the base (Separable algebraic elements and separable extensions).
Proof
For each , [L1] supplies its irreducible minimal polynomial over , and [L2] makes that polynomial separable.
Thus every element of is separable over , so is separable by [L3].
Depends on
Used by
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 19 results over 6 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)