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.
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- 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.
For prime and , the congruence has at most residue-class solutions
Statement
If is prime and , then
has at most distinct solution classes modulo .
Facts & Assumptions
Given: A prime and an integer .
The residue-class ring is a field when is prime (For every prime , the two operations on make it a field).
Every field is a commutative ring and has no zero divisors (Every field is a commutative ring with ; it is an integral domain, and it is a commutative division ring).
A nonzero polynomial of degree over an integral domain has at most roots (A nonzero polynomial of degree over an integral domain has at most distinct roots).
Congruence modulo is equality of classes in , and a root is an element at which polynomial evaluation is zero (The congruence class and the quotient set , Evaluation and roots of a polynomial in a commutative target ring).
Proof
By [L1] and [L2], is an integral domain. The polynomial over this domain is nonzero and has degree .
By [L4], the solution classes of the congruence are exactly the roots of in . The bound [L3] therefore gives at most such classes.
Depends on
- For every prime $p$, the two operations on $\mathbb{Z}/p$ make it a field
- Every field is a commutative ring with $1 \ne 0$; it is an integral domain, and it is a commutative division ring
- A nonzero polynomial of degree $n$ over an integral domain has at most $n$ distinct roots
- The congruence class $[a]_n$ and the quotient set $\mathbb{Z}/n$
- Evaluation and roots of a polynomial in a commutative target ring
Used by
Dependency tree · two levels
28 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
- Peter Hackman, Elementary Number Theory, Chapter C (standard reference, not scraped)