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ring_theory.chain_of_divisors

Chains of divisors #

The results in this file show that in the monoid associates M of a unique_factorization_monoid M, an element a is an n-th prime power iff its set of divisors is a strictly increasing chain of length n + 1, meaning that we can find a strictly increasing bijection between fin (n + 1) and the set of factors of a.

Main results #

Todo #

theorem divisor_chain.exists_chain_of_prime_pow {M : Type u_1} [cancel_comm_monoid_with_zero M] {p : associates M} {n : } (hn : n 0) (hp : prime p) :
∃ (c : fin (n + 1)associates M), c 1 = p strict_mono c ∀ {r : associates M}, r p ^ n ∃ (i : fin (n + 1)), r = c i
theorem divisor_chain.element_of_chain_not_is_unit_of_index_ne_zero {M : Type u_1} [cancel_comm_monoid_with_zero M] {n : } {i : fin (n + 1)} (i_pos : i 0) {c : fin (n + 1)associates M} (h₁ : strict_mono c) :
¬is_unit (c i)
theorem divisor_chain.first_of_chain_is_unit {M : Type u_1} [cancel_comm_monoid_with_zero M] {q : associates M} {n : } {c : fin (n + 1)associates M} (h₁ : strict_mono c) (h₂ : ∀ {r : associates M}, r q ∃ (i : fin (n + 1)), r = c i) :
is_unit (c 0)
theorem divisor_chain.second_of_chain_is_irreducible {M : Type u_1} [cancel_comm_monoid_with_zero M] {q : associates M} {n : } (hn : n 0) {c : fin (n + 1)associates M} (h₁ : strict_mono c) (h₂ : ∀ {r : associates M}, r q ∃ (i : fin (n + 1)), r = c i) (hq : q 0) :

The second element of a chain is irreducible.

theorem divisor_chain.eq_second_of_chain_of_prime_dvd {M : Type u_1} [cancel_comm_monoid_with_zero M] {p q r : associates M} {n : } (hn : n 0) {c : fin (n + 1)associates M} (h₁ : strict_mono c) (h₂ : ∀ {r : associates M}, r q ∃ (i : fin (n + 1)), r = c i) (hp : prime p) (hr : r q) (hp' : p r) :
p = c 1
theorem divisor_chain.card_subset_divisors_le_length_of_chain {M : Type u_1} [cancel_comm_monoid_with_zero M] {q : associates M} {n : } {c : fin (n + 1)associates M} (h₂ : ∀ {r : associates M}, r q ∃ (i : fin (n + 1)), r = c i) {m : finset (associates M)} (hm : ∀ (r : associates M), r mr q) :
m.card n + 1
theorem divisor_chain.element_of_chain_eq_pow_second_of_chain {M : Type u_1} [cancel_comm_monoid_with_zero M] [unique_factorization_monoid M] {q r : associates M} {n : } (hn : n 0) {c : fin (n + 1)associates M} (h₁ : strict_mono c) (h₂ : ∀ {r : associates M}, r q ∃ (i : fin (n + 1)), r = c i) (hr : r q) (hq : q 0) :
∃ (i : fin (n + 1)), r = c 1 ^ i
theorem divisor_chain.eq_pow_second_of_chain_of_has_chain {M : Type u_1} [cancel_comm_monoid_with_zero M] [unique_factorization_monoid M] {q : associates M} {n : } (hn : n 0) {c : fin (n + 1)associates M} (h₁ : strict_mono c) (h₂ : ∀ {r : associates M}, r q ∃ (i : fin (n + 1)), r = c i) (hq : q 0) :
q = c 1 ^ n
theorem divisor_chain.is_prime_pow_of_has_chain {M : Type u_1} [cancel_comm_monoid_with_zero M] [unique_factorization_monoid M] {q : associates M} {n : } (hn : n 0) {c : fin (n + 1)associates M} (h₁ : strict_mono c) (h₂ : ∀ {r : associates M}, r q ∃ (i : fin (n + 1)), r = c i) (hq : q 0) :
theorem pow_image_of_prime_by_factor_order_iso_dvd {M : Type u_1} [cancel_comm_monoid_with_zero M] {N : Type u_2} [cancel_comm_monoid_with_zero N] [unique_factorization_monoid N] [decidable_eq (associates M)] [unique_factorization_monoid M] {m p : associates M} {n : associates N} (hn : n 0) (hp : p unique_factorization_monoid.normalized_factors m) (d : {l // l m} ≃o {l // l n}) {s : } (hs' : p ^ s m) :
(d p, _⟩) ^ s n