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 splitting field of over is
Example
The splitting field over of the family is .
Facts & Assumptions
Given: The family .
The nonnegative real numbers and have square roots with the defining square equations (Square roots exist: a unique with ; the positives are ).
The splitting field of a product inside a common extension is the composite of the splitting fields of its two factors (Inside a common extension, the splitting field of is the composite of the splitting fields of and ).
A finite family has the same splitting field as the product of its nonzero members (Every finite family of nonzero polynomials has a splitting field, obtained from their product).
A splitting field is the field generated over the base by all roots of a polynomial that splits there (Polynomials that split and splitting fields of a polynomial or a family of polynomials).
Verification
By [F1], the roots of are , so [F4] makes its splitting field . Similarly the splitting field of is .
Their composite is the smallest field containing both, namely . By [F2] it splits the product, and by [F3] it is the splitting field of the stated family.
Depends on
- Every finite family of nonzero polynomials has a splitting field, obtained from their product
- Inside a common extension, the splitting field of $fg$ is the composite of the splitting fields of $f$ and $g$
- Polynomials that split and splitting fields of a polynomial or a family of polynomials
- Square roots exist: a unique $\sqrt{a} \ge 0$ with $(\sqrt{a})^2 = a$; the positives are $\{x^2 : x \neq 0\}$
- The rationals form a field
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
22 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
- J. S. Milne, Fields and Galois Theory, Chapter 2 (standard reference, not scraped)