mathlib documentation

ring_theory.unique_factorization_domain

Unique factorization #

Main Definitions #

To do #

@[class]
structure wf_dvd_monoid (α : Type u_2) [comm_monoid_with_zero α] :
Prop

Well-foundedness of the strict version of |, which is equivalent to the descending chain condition on divisibility and to the ascending chain condition on principal ideals in an integral domain.

Instances
@[protected, instance]
@[protected, instance]
theorem wf_dvd_monoid.exists_irreducible_factor {α : Type u_1} [comm_monoid_with_zero α] [wf_dvd_monoid α] {a : α} (ha : ¬is_unit a) (ha0 : a 0) :
∃ (i : α), irreducible i i a
theorem wf_dvd_monoid.induction_on_irreducible {α : Type u_1} [comm_monoid_with_zero α] [wf_dvd_monoid α] {P : α → Prop} (a : α) (h0 : P 0) (hu : ∀ (u : α), is_unit uP u) (hi : ∀ (a i : α), a 0irreducible iP aP (i * a)) :
P a
theorem wf_dvd_monoid.exists_factors {α : Type u_1} [comm_monoid_with_zero α] [wf_dvd_monoid α] (a : α) :
a 0(∃ (f : multiset α), (∀ (b : α), b firreducible b) associated f.prod a)
@[class]
structure unique_factorization_monoid (α : Type u_2) [cancel_comm_monoid_with_zero α] :
Prop

unique factorization monoids.

These are defined as cancel_comm_monoid_with_zeros with well-founded strict divisibility relations, but this is equivalent to more familiar definitions:

Each element (except zero) is uniquely represented as a multiset of irreducible factors. Uniqueness is only up to associated elements.

Each element (except zero) is non-uniquely represented as a multiset of prime factors.

To define a UFD using the definition in terms of multisets of irreducible factors, use the definition of_exists_unique_irreducible_factors

To define a UFD using the definition in terms of multisets of prime factors, use the definition of_exists_prime_factors

Instances

Can't be an instance because it would cause a loop ufm → wf_dvd_monoid → ufm → ....

theorem unique_factorization_monoid.exists_prime_factors {α : Type u_1} [cancel_comm_monoid_with_zero α] [unique_factorization_monoid α] (a : α) :
a 0(∃ (f : multiset α), (∀ (b : α), b fprime b) associated f.prod a)
theorem unique_factorization_monoid.induction_on_prime {α : Type u_1} [cancel_comm_monoid_with_zero α] [unique_factorization_monoid α] {P : α → Prop} (a : α) (h₁ : P 0) (h₂ : ∀ (x : α), is_unit xP x) (h₃ : ∀ (a p : α), a 0prime pP aP (p * a)) :
P a
theorem unique_factorization_monoid.factors_unique {α : Type u_1} [cancel_comm_monoid_with_zero α] [unique_factorization_monoid α] {f g : multiset α} :
(∀ (x : α), x firreducible x)(∀ (x : α), x girreducible x)associated f.prod g.prodmultiset.rel associated f g
theorem prime_factors_unique {α : Type u_1} [cancel_comm_monoid_with_zero α] {f g : multiset α} :
(∀ (x : α), x fprime x)(∀ (x : α), x gprime x)associated f.prod g.prodmultiset.rel associated f g
theorem prime_factors_irreducible {α : Type u_1} [cancel_comm_monoid_with_zero α] {a : α} {f : multiset α} (ha : irreducible a) (pfa : (∀ (b : α), b fprime b) associated f.prod a) :
∃ (p : α), associated a p f = {p}

If an irreducible has a prime factorization, then it is an associate of one of its prime factors.

theorem wf_dvd_monoid.of_exists_prime_factors {α : Type u_1} [cancel_comm_monoid_with_zero α] (pf : ∀ (a : α), a 0(∃ (f : multiset α), (∀ (b : α), b fprime b) associated f.prod a)) :
theorem irreducible_iff_prime_of_exists_prime_factors {α : Type u_1} [cancel_comm_monoid_with_zero α] (pf : ∀ (a : α), a 0(∃ (f : multiset α), (∀ (b : α), b fprime b) associated f.prod a)) {p : α} :
theorem unique_factorization_monoid.of_exists_prime_factors {α : Type u_1} [cancel_comm_monoid_with_zero α] (pf : ∀ (a : α), a 0(∃ (f : multiset α), (∀ (b : α), b fprime b) associated f.prod a)) :
theorem unique_factorization_monoid.iff_exists_prime_factors {α : Type u_1} [cancel_comm_monoid_with_zero α] :
unique_factorization_monoid α ∀ (a : α), a 0(∃ (f : multiset α), (∀ (b : α), b fprime b) associated f.prod a)
theorem irreducible_iff_prime_of_exists_unique_irreducible_factors {α : Type u_1} [cancel_comm_monoid_with_zero α] (eif : ∀ (a : α), a 0(∃ (f : multiset α), (∀ (b : α), b firreducible b) associated f.prod a)) (uif : ∀ (f g : multiset α), (∀ (x : α), x firreducible x)(∀ (x : α), x girreducible x)associated f.prod g.prodmultiset.rel associated f g) (p : α) :
theorem unique_factorization_monoid.of_exists_unique_irreducible_factors {α : Type u_1} [cancel_comm_monoid_with_zero α] (eif : ∀ (a : α), a 0(∃ (f : multiset α), (∀ (b : α), b firreducible b) associated f.prod a)) (uif : ∀ (f g : multiset α), (∀ (x : α), x firreducible x)(∀ (x : α), x girreducible x)associated f.prod g.prodmultiset.rel associated f g) :

Noncomputably determines the multiset of prime factors.

Equations
@[simp]

An arbitrary choice of factors of x : M is exactly the (unique) normalized set of factors, if M has a trivial group of units.

@[protected]

Noncomputably defines a normalization_monoid structure on a unique_factorization_monoid.

Equations
theorem unique_factorization_monoid.no_factors_of_no_prime_factors {R : Type u_2} [cancel_comm_monoid_with_zero R] [unique_factorization_monoid R] {a b : R} (ha : a 0) (h : ∀ {d : R}, d ad b¬prime d) {d : R} :
d ad bis_unit d
theorem unique_factorization_monoid.dvd_of_dvd_mul_left_of_no_prime_factors {R : Type u_2} [cancel_comm_monoid_with_zero R] [unique_factorization_monoid R] {a b c : R} (ha : a 0) :
(∀ {d : R}, d ad c¬prime d)a b * ca b

Euclid's lemma: if a ∣ b * c and a and c have no common prime factors, a ∣ b. Compare is_coprime.dvd_of_dvd_mul_left.

theorem unique_factorization_monoid.dvd_of_dvd_mul_right_of_no_prime_factors {R : Type u_2} [cancel_comm_monoid_with_zero R] [unique_factorization_monoid R] {a b c : R} (ha : a 0) (no_factors : ∀ {d : R}, d ad b¬prime d) :
a b * ca c

Euclid's lemma: if a ∣ b * c and a and b have no common prime factors, a ∣ c. Compare is_coprime.dvd_of_dvd_mul_right.

theorem unique_factorization_monoid.exists_reduced_factors {R : Type u_2} [cancel_comm_monoid_with_zero R] [unique_factorization_monoid R] (a : R) (H : a 0) (b : R) :
∃ (a' b' c' : R), (∀ {d : R}, d a'd b'is_unit d) c' * a' = a c' * b' = b

If a ≠ 0, b are elements of a unique factorization domain, then dividing out their common factor c' gives a' and b' with no factors in common.

theorem unique_factorization_monoid.exists_reduced_factors' {R : Type u_2} [cancel_comm_monoid_with_zero R] [unique_factorization_monoid R] (a b : R) (hb : b 0) :
∃ (a' b' c' : R), (∀ {d : R}, d a'd b'is_unit d) c' * a' = a c' * b' = b

The multiplicity of an irreducible factor of a nonzero element is exactly the number of times the normalized factor occurs in the normalized_factors.

See also count_normalized_factors_eq which expands the definition of multiplicity to produce a specification for count (normalized_factors _) _..

The number of times an irreducible factor p appears in normalized_factors x is defined by the number of times it divides x.

See also multiplicity_eq_count_normalized_factors if n is given by multiplicity p x.

def associates.factor_set (α : Type u) [cancel_comm_monoid_with_zero α] :
Type u

factor_set α representation elements of unique factorization domain as multisets. multiset α produced by normalized_factors are only unique up to associated elements, while the multisets in factor_set α are unique by equality and restricted to irreducible elements. This gives us a representation of each element as a unique multisets (or the added ⊤ for 0), which has a complete lattice struture. Infimum is the greatest common divisor and supremum is the least common multiple.

Equations
theorem associates.factor_set.coe_add {α : Type u_1} [cancel_comm_monoid_with_zero α] {a b : multiset {a // irreducible a}} :
(a + b) = a + b

Evaluates the product of a factor_set to be the product of the corresponding multiset, or 0 if there is none.

Equations
@[simp]
theorem associates.prod_top {α : Type u_1} [cancel_comm_monoid_with_zero α] :
@[simp]
theorem associates.prod_coe {α : Type u_1} [cancel_comm_monoid_with_zero α] {s : multiset {a // irreducible a}} :
@[simp]
theorem associates.prod_add {α : Type u_1} [cancel_comm_monoid_with_zero α] (a b : associates.factor_set α) :
(a + b).prod = (a.prod) * b.prod
theorem associates.prod_mono {α : Type u_1} [cancel_comm_monoid_with_zero α] {a b : associates.factor_set α} :
a ba.prod b.prod

bcount p s is the multiplicity of p in the factor_set s (with bundled p)

Equations
def associates.count {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [decidable_eq (associates α)] (p : associates α) :

count p s is the multiplicity of the irreducible p in the factor_set s.

If p is not irreducible, count p s is defined to be 0.

Equations
@[simp]
theorem associates.count_some {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [decidable_eq (associates α)] {p : associates α} (hp : irreducible p) (s : multiset {a // irreducible a}) :
p.count (some s) = multiset.count p, hp⟩ s
@[simp]
theorem associates.count_zero {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [decidable_eq (associates α)] {p : associates α} (hp : irreducible p) :
p.count 0 = 0
theorem associates.count_reducible {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [decidable_eq (associates α)] {p : associates α} (hp : ¬irreducible p) :
p.count = 0
def associates.bfactor_set_mem {α : Type u_1} [cancel_comm_monoid_with_zero α] :
{a // irreducible a}associates.factor_set α → Prop

membership in a factor_set (bundled version)

Equations
def associates.factor_set_mem {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] (p : associates α) (s : associates.factor_set α) :
Prop

factor_set_mem p s is the predicate that the irreducible p is a member of s : factor_set α.

If p is not irreducible, p is not a member of any factor_set.

Equations
@[protected, instance]
Equations
@[simp]
theorem associates.factor_set_mem_eq_mem {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] (p : associates α) (s : associates.factor_set α) :
p.factor_set_mem s = (p s)
theorem associates.mem_factor_set_top {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] {p : associates α} {hp : irreducible p} :
theorem associates.mem_factor_set_some {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] {p : associates α} {hp : irreducible p} {l : multiset {a // irreducible a}} :
p l p, hp⟩ l
theorem associates.reducible_not_mem_factor_set {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] {p : associates α} (hp : ¬irreducible p) (s : associates.factor_set α) :
p s
theorem associates.unique' {α : Type u_1} [cancel_comm_monoid_with_zero α] [unique_factorization_monoid α] {p q : multiset (associates α)} :
(∀ (a : associates α), a pirreducible a)(∀ (a : associates α), a qirreducible a)p.prod = q.prodp = q
theorem associates.prod_le_prod_iff_le {α : Type u_1} [cancel_comm_monoid_with_zero α] [unique_factorization_monoid α] [nontrivial α] {p q : multiset (associates α)} (hp : ∀ (a : associates α), a pirreducible a) (hq : ∀ (a : associates α), a qirreducible a) :
p.prod q.prod p q
noncomputable def associates.factors' {α : Type u_1} [cancel_comm_monoid_with_zero α] [unique_factorization_monoid α] [dec : decidable_eq α] (a : α) :

This returns the multiset of irreducible factors as a factor_set, a multiset of irreducible associates with_top.

Equations
noncomputable def associates.factors {α : Type u_1} [cancel_comm_monoid_with_zero α] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] (a : associates α) :

This returns the multiset of irreducible factors of an associate as a factor_set, a multiset of irreducible associates with_top.

Equations
@[simp]
@[simp]
theorem associates.eq_factors_of_eq_counts {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] {a b : associates α} (ha : a 0) (hb : b 0) (h : ∀ (p : associates α), irreducible pp.count a.factors = p.count b.factors) :
theorem associates.eq_of_eq_counts {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] {a b : associates α} (ha : a 0) (hb : b 0) (h : ∀ (p : associates α), irreducible pp.count a.factors = p.count b.factors) :
a = b
theorem associates.count_le_count_of_factors_le {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] {a b p : associates α} (hb : b 0) (hp : irreducible p) (h : a.factors b.factors) :
@[simp]
theorem associates.factors_mul {α : Type u_1} [cancel_comm_monoid_with_zero α] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] (a b : associates α) :
theorem associates.count_le_count_of_le {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] {a b p : associates α} (hb : b 0) (hp : irreducible p) (h : a b) :
@[protected, instance]
noncomputable def associates.has_sup {α : Type u_1} [cancel_comm_monoid_with_zero α] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] :
Equations
@[protected, instance]
noncomputable def associates.has_inf {α : Type u_1} [cancel_comm_monoid_with_zero α] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] :
Equations
theorem associates.sup_mul_inf {α : Type u_1} [cancel_comm_monoid_with_zero α] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] (a b : associates α) :
(a b) * (a b) = a * b
theorem associates.dvd_of_mem_factors {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] {a p : associates α} {hp : irreducible p} (hm : p a.factors) :
p a
theorem associates.dvd_of_mem_factors' {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] {a : α} {p : associates α} {hp : irreducible p} {hz : a 0} (h_mem : p, hp⟩ associates.factors' a) :
theorem associates.mem_factors'_of_dvd {α : Type u_1} [cancel_comm_monoid_with_zero α] [unique_factorization_monoid α] [dec : decidable_eq α] {a p : α} (ha0 : a 0) (hp : irreducible p) (hd : p a) :
theorem associates.mem_factors'_iff_dvd {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] {a p : α} (ha0 : a 0) (hp : irreducible p) :
theorem associates.mem_factors_of_dvd {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] {a p : α} (ha0 : a 0) (hp : irreducible p) (hd : p a) :
theorem associates.mem_factors_iff_dvd {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] {a p : α} (ha0 : a 0) (hp : irreducible p) :
theorem associates.exists_prime_dvd_of_not_inf_one {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] {a b : α} (ha : a 0) (hb : b 0) (h : associates.mk a associates.mk b 1) :
∃ (p : α), prime p p a p b
theorem associates.coprime_iff_inf_one {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] {a b : α} (ha0 : a 0) (hb0 : b 0) :
associates.mk a associates.mk b = 1 ∀ {d : α}, d ad b¬prime d
theorem associates.factors_self {α : Type u_1} [cancel_comm_monoid_with_zero α] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] [nontrivial α] {p : associates α} (hp : irreducible p) :
p.factors = some {p, hp⟩}
theorem associates.factors_prime_pow {α : Type u_1} [cancel_comm_monoid_with_zero α] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] [nontrivial α] {p : associates α} (hp : irreducible p) (k : ) :
(p ^ k).factors = some (multiset.repeat p, hp⟩ k)
theorem associates.prime_pow_dvd_iff_le {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] [nontrivial α] {m p : associates α} (h₁ : m 0) (h₂ : irreducible p) {k : } :
p ^ k m k p.count m.factors
theorem associates.le_of_count_ne_zero {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] {m p : associates α} (h0 : m 0) (hp : irreducible p) :
p.count m.factors 0p m
theorem associates.count_ne_zero_iff_dvd {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] {a p : α} (ha0 : a 0) (hp : irreducible p) :
theorem associates.count_self {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] [nontrivial α] {p : associates α} (hp : irreducible p) :
theorem associates.count_eq_zero_of_ne {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] {p q : associates α} (hp : irreducible p) (hq : irreducible q) (h : p q) :
theorem associates.count_mul {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] {a : associates α} (ha : a 0) {b : associates α} (hb : b 0) {p : associates α} (hp : irreducible p) :
theorem associates.count_of_coprime {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] {a : associates α} (ha : a 0) {b : associates α} (hb : b 0) (hab : ∀ (d : associates α), d ad b¬prime d) {p : associates α} (hp : irreducible p) :
theorem associates.count_mul_of_coprime {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] {a b : associates α} (hb : b 0) {p : associates α} (hp : irreducible p) (hab : ∀ (d : associates α), d ad b¬prime d) :
theorem associates.count_mul_of_coprime' {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] {a b p : associates α} (hp : irreducible p) (hab : ∀ (d : associates α), d ad b¬prime d) :
theorem associates.dvd_count_of_dvd_count_mul {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] {a b : associates α} (hb : b 0) {p : associates α} (hp : irreducible p) (hab : ∀ (d : associates α), d ad b¬prime d) {k : } (habk : k p.count (a * b).factors) :
@[simp]
@[simp]
theorem associates.pow_factors {α : Type u_1} [cancel_comm_monoid_with_zero α] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] [nontrivial α] {a : associates α} {k : } :
(a ^ k).factors = k a.factors
theorem associates.count_pow {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] [nontrivial α] {a : associates α} (ha : a 0) {p : associates α} (hp : irreducible p) (k : ) :
p.count (a ^ k).factors = k * p.count a.factors
theorem associates.dvd_count_pow {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] [nontrivial α] {a : associates α} (ha : a 0) {p : associates α} (hp : irreducible p) (k : ) :
k p.count (a ^ k).factors
theorem associates.is_pow_of_dvd_count {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] [nontrivial α] {a : associates α} (ha : a 0) {k : } (hk : ∀ (p : associates α), irreducible pk p.count a.factors) :
∃ (b : associates α), a = b ^ k
theorem associates.eq_pow_count_factors_of_dvd_pow {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] {p a : associates α} (hp : irreducible p) {n : } (h : a p ^ n) :
a = p ^ p.count a.factors

The only divisors of prime powers are prime powers. See eq_pow_find_of_dvd_irreducible_pow for an explicit expression as a p-power (without using count).

theorem associates.count_factors_eq_find_of_dvd_pow {α : Type u_1} [cancel_comm_monoid_with_zero α] [dec_irr : Π (p : associates α), decidable (irreducible p)] [unique_factorization_monoid α] [dec : decidable_eq α] [dec' : decidable_eq (associates α)] {a p : associates α} (hp : irreducible p) [Π (n : ), decidable (a p ^ n)] {n : } (h : a p ^ n) :
theorem associates.eq_pow_of_mul_eq_pow {α : Type u_1} [cancel_comm_monoid_with_zero α] [unique_factorization_monoid α] [nontrivial α] {a b c : associates α} (ha : a 0) (hb : b 0) (hab : ∀ (d : associates α), d ad b¬prime d) {k : } (h : a * b = c ^ k) :
∃ (d : associates α), a = d ^ k
theorem associates.eq_pow_find_of_dvd_irreducible_pow {α : Type u_1} [cancel_comm_monoid_with_zero α] [unique_factorization_monoid α] {a p : associates α} (hp : irreducible p) [Π (n : ), decidable (a p ^ n)] {n : } (h : a p ^ n) :
a = p ^ nat.find _

The only divisors of prime powers are prime powers.

theorem associates.quot_out {α : Type u_1} [comm_monoid α] (a : associates α) :

to_gcd_monoid constructs a GCD monoid out of a unique factorization domain.

Equations

to_normalized_gcd_monoid constructs a GCD monoid out of a normalization on a unique factorization domain.

Equations
noncomputable def unique_factorization_monoid.fintype_subtype_dvd {M : Type u_1} [cancel_comm_monoid_with_zero M] [unique_factorization_monoid M] [fintype Mˣ] (y : M) (hy : y 0) :
fintype {x // x y}

If y is a nonzero element of a unique factorization monoid with finitely many units (e.g. , ideal (ring_of_integers K)), it has finitely many divisors.

Equations
noncomputable def factorization {α : Type u_1} [cancel_comm_monoid_with_zero α] [unique_factorization_monoid α] [normalization_monoid α] [decidable_eq α] (n : α) :

This returns the multiset of irreducible factors as a finsupp

Equations
@[simp]

The support of factorization n is exactly the finset of normalized factors

@[simp]
theorem factorization_mul {α : Type u_1} [cancel_comm_monoid_with_zero α] [unique_factorization_monoid α] [normalization_monoid α] [decidable_eq α] {a b : α} (ha : a 0) (hb : b 0) :

For nonzero a and b, the power of p in a * b is the sum of the powers in a and b

For any p, the power of p in x^n is n times the power in x