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.
Every prime modulus admits a primitive root
Statement
Every prime admits a primitive root modulo .
Facts & Assumptions
Given: A prime .
The unit group is a finite cyclic group (For every prime , the multiplicative group is cyclic).
A unit is a primitive root exactly when it generates the unit group (A unit is a primitive root modulo if and only if it generates ).
Proof
Choose a generator of the cyclic group in [L1]; such a generator exists also when , since the one-element group is cyclic.
By [L2], the chosen is a primitive root modulo .
Depends on
Used by
- A coprime exponent gives a unique nonzero k-th root modulo a prime Corollary
- x²≡ a (mod p) has exactly 1+(a/p) solution classes Corollary
- Fourth and eighth powers modulo 17 Example
- An odd prime has (p-1)/2 nonzero quadratic residues and as many nonresidues Theorem
- Euler's criterion: (a/p)≡ a^(p-1)/2 (mod p) Theorem
- For every odd prime p and k≥1, (ℤ/pᵏℤ)^× is cyclic of order pᵏ⁻¹(p-1) Theorem
Dependency tree · two levels
10 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, Theorem C.II.1 (standard reference, not scraped)
- William Stein, Elementary Number Theory, Theorem 2.5.8 (standard reference, not scraped)