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
- For every finite field F_q and every n≥1, a monic irreducible polynomial of degree n exists Corollary
- A normal algebraic extension is one in which every minimal polynomial with a root in the extension splits there Definition
- An algebraic closure of a field Definition
- Purely inseparable algebraic extensions Definition
- Separable algebraic elements and separable extensions Definition
- 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
- Algebraicity is transitive in towers of field extensions Theorem
- An extension generated by finitely many algebraic elements is finite Theorem
- Every finite field extension is algebraic Theorem
- The elements of an extension algebraic over the base field form a subfield 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 · two levels
8 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
- T. Judson, Abstract Algebra: Theory and Applications, Extension Fields (standard reference, not scraped)