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ring_theory.​euclidean_domain

ring_theory.​euclidean_domain

source
dvd_or_coprime
gcd_is_unit_iff
is_coprime_of_dvd
span_gcd
source
theorem span_gcd {α : Type u_1} [euclidean_domain α] (x y : α) :
ideal.span {euclidean_domain.gcd x y} = ideal.span {x, y}

source
theorem gcd_is_unit_iff {α : Type u_1} [euclidean_domain α] {x y : α} :
is_unit (euclidean_domain.gcd x y) ↔ is_coprime x y

source
theorem is_coprime_of_dvd {α : Type u_1} [euclidean_domain α] {x y : α} :
¬(x = 0 ∧ y = 0) → (∀ (z : α), z ∈ nonunits α → z ≠ 0 → z ∣ x → ¬z ∣ y) → is_coprime x y

source
theorem dvd_or_coprime {α : Type u_1} [euclidean_domain α] (x y : α) :
irreducible x → x ∣ y ∨ is_coprime x y

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