The integers have no zero divisors; multiplicative cancellation

Depends on

• Arithmetic on the integers
• The integers form a commutative ring
• The integers form a totally ordered ring

Used by

• A rational root of xᵏ = m is an integer: if k ≥ 1, m ∈ ℤ, x ∈ ℚ and xᵏ is the image of m, then x is the image of an integer Corollary
• Every nonzero integer n is u ∏_i<r pᵢ with u ∈ {1,-1} and every pᵢ prime; u and r are determined by n, and the list is determined up to a permutation Corollary
• For an integer p > 1: p is prime if and only if, for all integers a and b, p ∣ ab implies p ∣ a or p ∣ b Corollary
• For every prime p and positive n, xⁿ-p is irreducible over ℚ Corollary
• If d = gcd(a,b) is nonzero then a/d and b/d are coprime Corollary
• 2ℤ has index 2 in ℤ and is nevertheless equinumerous with ℤ Counterexample
• 2ℤ is closed under addition, negation and multiplication and is not a subring of ℤ, because it does not contain 1 Counterexample
• In the multiplicative monoid H = {1, 4, 7, 10, …} of positive integers one more than a multiple of 3, the element 100 has two genuinely different factorisations into irreducibles, 4 · 25 and 10 · 10 Counterexample
• The zero ideal of ℤ is prime but not maximal Counterexample
• Arithmetic on the rationals Definition
• Common multiple, and the least common multiple lcm(a,b), taken to be 0 when a = 0 or b = 0 Definition
• 12ℤ + 18ℤ = 6ℤ and 12ℤ ∩ 18ℤ = 36ℤ, the arithmetic of gcd and lcm read off the subgroups of (ℤ,+) Example
• Bézout coefficients are not unique: 1071 · (-3) + 462 · 7 = 21 and 1071 · 19 + 462 · (-44) = 21, and for nonzero a, b every solution has the form (x₀ + t b/d, y₀ - t a/d) Example
• Every positive divisor of the order of a finite cyclic group occurs as the order of a subgroup Example
• No rational squares to 3 or to 6, and none cubes to 2: three instances of the rational-root corollary Example
• The 2-adic absolute value gives an ultrametric on ℚ, in which every triangle is isosceles and every point of a ball is a centre Example
• The integers are a Euclidean domain with Euclidean function δ(b)=|b| Example
• ℤ is an integral domain of characteristic 0 whose group of units is {1,-1}, so it is not a field: 2 is nonzero and not invertible Example
• FALSE: For all integers a and b, gcd(a,b) · lcm(a,b) = ab False statement
• FALSE: for every finite list p₀, …, pₙ₋₁ of distinct primes, p₀ ⋯ pₙ₋₁ + 1 is prime False statement
• FALSE: In every commutative ring, each nonzero element is either a unit or a zero divisor False statement
• For a prime p and a nonzero integer a: p^vₚ(a) ∣ a and p^vₚ(a)+1 ∤ a; pᵏ ∣ a holds exactly for k ≤ vₚ(a); vₚ(a) ≥ 1 exactly when p ∣ a; vₚ(1) = vₚ(-1) = 0; and vₚ(p) = 1 Lemma
• For a prime p and k≥1, multiplication by p bijects the standard representatives modulo pᵏ⁻¹ with the representatives modulo pᵏ divisible by p Lemma
• For coprime positive m,n, every pair of residue classes modulo m and n has a unique simultaneous class modulo mn Lemma
• For positive integers a and b: a ∣ b if and only if vₚ(a) ≤ vₚ(b) for every prime p Lemma
• gcd(ca, cb) = |c| gcd(a,b) for all integers a, b, c, the identity holding at c = 0 and at (a,b) = (0,0) as well Lemma
• If n≥1, d=gcd(a,n) and d∣ b, then ax≡ bpmod n reduces to a congruence with coprime coefficient modulo n/d Lemma
• The p-adic valuation extends to the nonzero rationals by vₚ(a/b) := vₚ(a) - vₚ(b) ∈ ℤ, independently of the representation; it satisfies vₚ(xy) = vₚ(x) + vₚ(y), and vₚ(x+y) ≥ min{vₚ(x), vₚ(y)} whenever x, y and x+y are nonzero Lemma
• The rational-defining relation is an equivalence relation Lemma
• vₚ(ab) = vₚ(a) + vₚ(b) for nonzero integers a, b, and vₚ(a+b) ≥ min{vₚ(a), vₚ(b)} whenever a, b and a+b are all nonzero Lemma
• Euler's product formula φ(n)=n∏_p∣ n(1-1/p)=∏_pᵏ∥ n(pᵏ-pᵏ⁻¹) for n≥1, stated through a finite injective list of its prime divisors Theorem
• Every common multiple of a and b is a multiple of lcm(a,b), and gcd(a,b) · lcm(a,b) = |ab| Theorem
• For every positive integer n, ∑_d∣ n, d>0φ(d)=n Theorem
• For positive integers a and b and every prime p: vₚ(gcd(a,b)) = min{vₚ(a), vₚ(b)} and vₚ(lcm(a,b)) = max{vₚ(a), vₚ(b)}; so the exponent-wise greatest common divisor is the gcd of the divisibility page and not a second notion Theorem
• Gauss lemma: primitive factorisations over ℚ can be cleared to primitive factorisations over ℤ Theorem
• The fundamental theorem of arithmetic: every integer n ≥ 1 is a product of primes, and the factorisation is unique up to order — if ∏_i<r pᵢ = ∏_j<s qⱼ with every pᵢ and qⱼ prime, then r = s and qᵢ = p_π(i) for some π ∈ Sym(r) Theorem
• The rationals form a field Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 31 results over 12 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

• T. Tao, Analysis I, 3rd ed., §4.1 (standard reference, not scraped)
• Integer — construction from pairs of naturals (Wikipedia) (standard reference, not scraped)