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.
Spec Z has one generic point and closed prime-number points
Example
The points of are and for rational primes . The point is generic, while each is closed.
Facts & Assumptions
Given: The ring of integers .
is a field when is a rational prime (For every prime , the two operations on make it a field).
A quotient is a domain exactly when the defining ideal is prime ( is an integral domain if and only if is a prime ideal).
Verification
A nonzero prime ideal has a least positive member and division shows it is for a rational prime ; conversely [F1] and [F2] make every prime, while is prime because is a domain.
The ideals are maximal and therefore are closed points.
The closure of is , so it is generic.
These are exactly the claimed generic and closed points.
Depends on
- Closed points of an affine scheme
- Generic points of irreducible closed subsets
- Every irreducible closed subset of an affine spectrum has a unique generic point
- The well-ordering principle
- Division with remainder in $\mathbb{Z}$: for $a \in \mathbb{Z}$ and $b > 0$ there are unique $q, r \in \mathbb{Z}$ with $a = qb + r$ and $0 \le r < b$
- $R/P$ is an integral domain if and only if $P$ is a prime ideal
- For every prime $p$, the two operations on $\mathbb{Z}/p$ make it a field
- The integers form a commutative ring
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
47 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
- James S. Milne, Algebraic Geometry, 10.24 (standard reference, not scraped)