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.
Algebraically constructible real numbers as the smallest real subfield closed under positive square roots
Definition
Let be the intersection of all subfields such that and
This family is nonempty because belongs to it, and the intersection is again a subfield with the same square-root closure. A real number is algebraically constructible when it belongs to .
This is an algebraic definition. It does not assert an equivalence with a physical straightedge-and-compass model.
Depends on
- The rationals form a field
- The reals form a totally ordered field
- Square roots exist: a unique $\sqrt{a} \ge 0$ with $(\sqrt{a})^2 = a$; the positives are $\{x^2 : x \neq 0\}$
- Subfield: a subring of a field closed under inverses of its nonzero elements, and therefore a field with the restricted operations
Used by
Dependency tree · two levels
27 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
- A. W. Knapp, Basic Algebra, 2nd ed., Chapter IX, Section 5 (standard reference, not scraped)
- J. S. Milne, Fields and Galois Theory, Theorem 1.37 through consequence 1.41 (standard reference, not scraped)