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 every prime , the multiplicative group is cyclic
Statement
For every prime , the group is cyclic of order .
Facts & Assumptions
Given: A prime .
is a field (For every prime , the two operations on make it a field).
Every field is an integral domain (Every field is a commutative ring with ; it is an integral domain, and it is a commutative division ring).
Every finite subgroup of the unit group of an integral domain is cyclic (Every finite subgroup of the unit group of an integral domain is cyclic).
is finite of order (The unit group and Euler's totient for ).
For prime , (, and for every prime ).
Proof
By [L1] and [L2], is an integral domain.
By [L4], its entire unit group is a finite subgroup of its units, so [L3] makes it cyclic; [L4] and [L5] give its order. This includes , when the group is trivial.
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
- Every finite subgroup of the unit group of an integral domain is cyclic
- The unit group $(\mathbb{Z}/n)^\times$ and Euler's totient $\varphi(n)=\lvert(\mathbb{Z}/n)^\times\rvert$ for $n\ge1$
- $\varphi(1)=1$, and $\varphi(p)=p-1$ for every prime $p$
Used by
- Every prime modulus admits a primitive root Corollary
- For prime p and d≥1, the congruence xᵈ≡1 (mod p) has gcd(d,p-1) nonzero solutions Corollary
- On the units, the Legendre symbol is the unique nontrivial homomorphism to {±1} Proposition
- An odd prime has (p-1)/2 nonzero quadratic residues and as many nonresidues Theorem
- The nonzero squares modulo an odd prime form an index-two subgroup Theorem
Dependency tree · two levels
35 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
- William Stein, Elementary Number Theory, Theorem 2.5.8 (standard reference, not scraped)