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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 m r 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 factor_order_iso_map_one_eq_bot {M : Type u_1} [cancel_comm_monoid_with_zero M] {N : Type u_2} [cancel_comm_monoid_with_zero N] {m : associates M} {n : associates N} (d : {l // l m} ≃o {l // l n}) :
(d 1, _⟩) = 1
theorem coe_factor_order_iso_map_eq_one_iff {M : Type u_1} [cancel_comm_monoid_with_zero M] {N : Type u_2} [cancel_comm_monoid_with_zero N] {m u : associates M} {n : associates N} (hu' : u m) (d : (set.Iic m) ≃o (set.Iic n)) :
(d u, hu'⟩) = 1 u = 1
def mk_factor_order_iso_of_factor_dvd_equiv {M : Type u_1} [cancel_comm_monoid_with_zero M] {N : Type u_2} [cancel_comm_monoid_with_zero N] [unique Mˣ] [unique Nˣ] {m : M} {n : N} {d : {l // l m} {l // l n}} (hd : (l l' : {l // l m}), (d l) (d l') l l') :

The order isomorphism between the factors of mk m and the factors of mk n induced by a bijection between the factors of m and the factors of n that preserves .

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