How 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.
For every prime , the two operations on make it a field
Statement
For every prime , the operations of addition and multiplication on make it a field (Field).
Facts & Assumptions
Given: A prime integer .
Modular addition makes an abelian group, modular multiplication makes it a commutative monoid, and both distributive laws hold (For every natural , is an abelian group, multiplication is a commutative monoid operation, and both distributive laws hold).
For positive modulus , is invertible exactly when (For , is a unit if and only if ).
Equality is equivalent to , which means (The congruence class and the quotient set , Congruence modulo an integer: when , including the moduli and ).
A field is a commutative ring with in which every nonzero element is multiplicatively invertible (Field).
Proof
By [L1], all additive, multiplicative, identity, inverse, commutativity, associativity and distributive field axioms already hold except possibly and invertibility of nonzero classes.
Since , it does not divide , so [L4] gives .
If , then by [L4]. Hence by [L2], and [L3] makes invertible.
Steps 1.1 through 1.3 verify every clause of [L5], so is a field.
Depends on
- For every natural $n$, $(\mathbb{Z}/n,+)$ is an abelian group, multiplication is a commutative monoid operation, and both distributive laws hold
- For $n\ge1$, $[a]_n$ is a unit if and only if $\gcd(a,n)=1$
- For a prime $p$ and any integer $a$, $\gcd(p,a)$ is $p$ when $p \mid a$ and $1$ otherwise; so $p \nmid a$ makes $p$ and $a$ coprime
- Prime and composite integers: $p$ is prime when $p > 1$ and its only positive divisors are $1$ and $p$
- Field
- The congruence class $[a]_n$ and the quotient set $\mathbb{Z}/n$
- Congruence modulo an integer: $a\equiv b\pmod n$ when $n\mid(a-b)$, including the moduli $0$ and $1$
Used by
- Over F₂, the equation x+y=0 has exactly two solutions, so the infinite-field hypothesis is necessary Counterexample
- Over Fₚ, xᵖ-x and the zero polynomial induce the same function but are distinct polynomials Counterexample
- Over ℤ/2, an antisymmetric bilinear form need not be alternating Counterexample
- For every integer n>1, nℤ is a maximal ideal of ℤ if and only if n is prime Example
- In characteristic 2, x²+1=(x+1)² has zero derivative and a repeated root Example
- Reduction modulo 2 proves x³+17x+391 irreducible over ℚ Example
- In (ℤ/p)^×, inversion pairs every class except [1]ₚ and [-1]ₚ, which are the only self-inverse classes Lemma
- The product of primitive integer polynomials is primitive, and contents multiply Lemma
- Eisenstein criterion over the integers Theorem
- Irreducibility after reduction modulo a prime implies irreducibility over ℚ when the leading coefficient survives Theorem
- Wilson's theorem: for every prime p, (p-1)!≡-1pmod p Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 79 results over 25 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.
Sources
- K. Conrad, Modular Arithmetic (standard reference, not scraped)
- MIT 18.783 Lecture Notes 3 (standard reference, not scraped)