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.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
The base field is the splitting field of the empty family and of every nonzero constant polynomial
Example
For every field , the splitting field over of the empty family is . The same is true for every nonzero constant polynomial .
Facts & Assumptions
Given: A field .
A nonzero degree-zero polynomial splits as its leading coefficient times an empty product, and a splitting field is generated by the roots of the polynomial or family (Polynomials that split and splitting fields of a polynomial or a family of polynomials).
Verification
The empty family has no roots, and a nonzero constant also has no roots. In either case the subfield generated over by the empty root set is itself.
The empty family is vacuously split over , while is the required empty linear factorisation. Thus both cases satisfy the splitting-field definition.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 10 results over 4 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
- T. Judson, Abstract Algebra: Theory and Applications, Section 21.2 (standard reference, not scraped)