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.
Algebraic and transcendental elements and algebraic extensions
Definition
Let be a field extension (Field extensions, generated subrings , generated subfields , and simple extensions) and let . The element is algebraic over if for some nonzero polynomial (Evaluation and roots of a polynomial in a commutative target ring); it is transcendental over otherwise. The extension is algebraic if every element of is algebraic over , and transcendental otherwise.
Depends on
Used by
- ℂ/ℝ has power basis 1,i and degree 2 Corollary
- ℚ(√2)≅ℚ[x]/(x²-2) with basis 1,√2 Example
- The minimal polynomial of √2+√3 over ℚ is x⁴-4x²+1 Example
- A simple transcendental extension consists exactly of rational expressions in its generator Theorem
- The evaluation kernel and the unique monic irreducible minimal polynomial of an algebraic element Theorem
- Two simple transcendental extensions are uniquely F-isomorphic once their generators are matched Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 22 results over 9 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, Extension Fields (standard reference, not scraped)