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/- | |
Copyright (c) 2020 Johan Commelin. All rights reserved. | |
Released under Apache 2.0 license as described in the file LICENSE. | |
Authors: Johan Commelin, Chris Hughes | |
-/ | |
import data.fintype.card | |
import data.polynomial.ring_division | |
import group_theory.specific_groups.cyclic | |
import algebra.geom_sum | |
/-! | |
# Integral domains | |
Assorted theorems about integral domains. | |
## Main theorems | |
* `is_cyclic_of_subgroup_is_domain`: A finite subgroup of the units of an integral domain is cyclic. | |
* `fintype.field_of_domain`: A finite integral domain is a field. | |
## TODO | |
Prove Wedderburn's little theorem, which shows that all finite division rings are actually fields. | |
## Tags | |
integral domain, finite integral domain, finite field | |
-/ | |
section | |
open finset polynomial function | |
open_locale big_operators nat | |
section cancel_monoid_with_zero | |
-- There doesn't seem to be a better home for these right now | |
variables {M : Type*} [cancel_monoid_with_zero M] [fintype M] | |
lemma mul_right_bijective_of_fintype₀ {a : M} (ha : a ≠ 0) : bijective (λ b, a * b) := | |
fintype.injective_iff_bijective.1 $ mul_right_injective₀ ha | |
lemma mul_left_bijective_of_fintype₀ {a : M} (ha : a ≠ 0) : bijective (λ b, b * a) := | |
fintype.injective_iff_bijective.1 $ mul_left_injective₀ ha | |
/-- Every finite nontrivial cancel_monoid_with_zero is a group_with_zero. -/ | |
def fintype.group_with_zero_of_cancel (M : Type*) [cancel_monoid_with_zero M] [decidable_eq M] | |
[fintype M] [nontrivial M] : group_with_zero M := | |
{ inv := λ a, if h : a = 0 then 0 else fintype.bij_inv (mul_right_bijective_of_fintype₀ h) 1, | |
mul_inv_cancel := λ a ha, | |
by { simp [has_inv.inv, dif_neg ha], exact fintype.right_inverse_bij_inv _ _ }, | |
inv_zero := by { simp [has_inv.inv, dif_pos rfl] }, | |
..‹nontrivial M›, | |
..‹cancel_monoid_with_zero M› } | |
end cancel_monoid_with_zero | |
variables {R : Type*} {G : Type*} | |
section ring | |
variables [ring R] [is_domain R] [fintype R] | |
/-- Every finite domain is a division ring. | |
TODO: Prove Wedderburn's little theorem, | |
which shows a finite domain is in fact commutative, hence a field. -/ | |
def fintype.division_ring_of_is_domain (R : Type*) [ring R] [is_domain R] [decidable_eq R] | |
[fintype R] : division_ring R := | |
{ ..show group_with_zero R, from fintype.group_with_zero_of_cancel R, | |
..‹ring R› } | |
/-- Every finite commutative domain is a field. | |
TODO: Prove Wedderburn's little theorem, which shows a finite domain is automatically commutative, | |
dropping one assumption from this theorem. -/ | |
def fintype.field_of_domain (R) [comm_ring R] [is_domain R] [decidable_eq R] [fintype R] : | |
field R := | |
{ .. fintype.group_with_zero_of_cancel R, | |
.. ‹comm_ring R› } | |
lemma fintype.is_field_of_domain (R) [comm_ring R] [is_domain R] [fintype R] : | |
is_field R := @field.to_is_field R $ @@fintype.field_of_domain R _ _ (classical.dec_eq R) _ | |
end ring | |
variables [comm_ring R] [is_domain R] [group G] [fintype G] | |
lemma card_nth_roots_subgroup_units (f : G →* R) (hf : injective f) {n : ℕ} (hn : 0 < n) (g₀ : G) : | |
({g ∈ univ | g ^ n = g₀} : finset G).card ≤ (nth_roots n (f g₀)).card := | |
begin | |
haveI : decidable_eq R := classical.dec_eq _, | |
refine le_trans _ (nth_roots n (f g₀)).to_finset_card_le, | |
apply card_le_card_of_inj_on f, | |
{ intros g hg, | |
rw [sep_def, mem_filter] at hg, | |
rw [multiset.mem_to_finset, mem_nth_roots hn, ← f.map_pow, hg.2] }, | |
{ intros, apply hf, assumption } | |
end | |
/-- A finite subgroup of the unit group of an integral domain is cyclic. -/ | |
lemma is_cyclic_of_subgroup_is_domain (f : G →* R) (hf : injective f) : is_cyclic G := | |
begin | |
classical, | |
apply is_cyclic_of_card_pow_eq_one_le, | |
intros n hn, | |
convert (le_trans (card_nth_roots_subgroup_units f hf hn 1) (card_nth_roots n (f 1))) | |
end | |
/-- The unit group of a finite integral domain is cyclic. | |
To support `ℤˣ` and other infinite monoids with finite groups of units, this requires only | |
`fintype Rˣ` rather than deducing it from `fintype R`. -/ | |
instance [fintype Rˣ] : is_cyclic Rˣ := | |
is_cyclic_of_subgroup_is_domain (units.coe_hom R) $ units.ext | |
section | |
variables (S : subgroup Rˣ) [fintype S] | |
/-- A finite subgroup of the units of an integral domain is cyclic. -/ | |
instance subgroup_units_cyclic : is_cyclic S := | |
begin | |
refine is_cyclic_of_subgroup_is_domain ⟨(coe : S → R), _, _⟩ | |
(units.ext.comp subtype.val_injective), | |
{ simp }, | |
{ intros, simp }, | |
end | |
end | |
lemma card_fiber_eq_of_mem_range {H : Type*} [group H] [decidable_eq H] | |
(f : G →* H) {x y : H} (hx : x ∈ set.range f) (hy : y ∈ set.range f) : | |
(univ.filter $ λ g, f g = x).card = (univ.filter $ λ g, f g = y).card := | |
begin | |
rcases hx with ⟨x, rfl⟩, | |
rcases hy with ⟨y, rfl⟩, | |
refine card_congr (λ g _, g * x⁻¹ * y) _ _ (λ g hg, ⟨g * y⁻¹ * x, _⟩), | |
{ simp only [mem_filter, one_mul, monoid_hom.map_mul, mem_univ, mul_right_inv, | |
eq_self_iff_true, monoid_hom.map_mul_inv, and_self, forall_true_iff] {contextual := tt} }, | |
{ simp only [mul_left_inj, imp_self, forall_2_true_iff], }, | |
{ simp only [true_and, mem_filter, mem_univ] at hg, | |
simp only [hg, mem_filter, one_mul, monoid_hom.map_mul, mem_univ, mul_right_inv, | |
eq_self_iff_true, exists_prop_of_true, monoid_hom.map_mul_inv, and_self, | |
mul_inv_cancel_right, inv_mul_cancel_right], } | |
end | |
/-- In an integral domain, a sum indexed by a nontrivial homomorphism from a finite group is zero. | |
-/ | |
lemma sum_hom_units_eq_zero (f : G →* R) (hf : f ≠ 1) : ∑ g : G, f g = 0 := | |
begin | |
classical, | |
obtain ⟨x, hx⟩ : ∃ x : monoid_hom.range f.to_hom_units, | |
∀ y : monoid_hom.range f.to_hom_units, y ∈ submonoid.powers x, | |
from is_cyclic.exists_monoid_generator, | |
have hx1 : x ≠ 1, | |
{ rintro rfl, | |
apply hf, | |
ext g, | |
rw [monoid_hom.one_apply], | |
cases hx ⟨f.to_hom_units g, g, rfl⟩ with n hn, | |
rwa [subtype.ext_iff, units.ext_iff, subtype.coe_mk, monoid_hom.coe_to_hom_units, | |
one_pow, eq_comm] at hn, }, | |
replace hx1 : (x : R) - 1 ≠ 0, | |
from λ h, hx1 (subtype.eq (units.ext (sub_eq_zero.1 h))), | |
let c := (univ.filter (λ g, f.to_hom_units g = 1)).card, | |
calc ∑ g : G, f g | |
= ∑ g : G, f.to_hom_units g : rfl | |
... = ∑ u : Rˣ in univ.image f.to_hom_units, | |
(univ.filter (λ g, f.to_hom_units g = u)).card • u : sum_comp (coe : Rˣ → R) f.to_hom_units | |
... = ∑ u : Rˣ in univ.image f.to_hom_units, c • u : | |
sum_congr rfl (λ u hu, congr_arg2 _ _ rfl) -- remaining goal 1, proven below | |
... = ∑ b : monoid_hom.range f.to_hom_units, c • ↑b : finset.sum_subtype _ | |
(by simp ) _ | |
... = c • ∑ b : monoid_hom.range f.to_hom_units, (b : R) : smul_sum.symm | |
... = c • 0 : congr_arg2 _ rfl _ -- remaining goal 2, proven below | |
... = 0 : smul_zero _, | |
{ -- remaining goal 1 | |
show (univ.filter (λ (g : G), f.to_hom_units g = u)).card = c, | |
apply card_fiber_eq_of_mem_range f.to_hom_units, | |
{ simpa only [mem_image, mem_univ, exists_prop_of_true, set.mem_range] using hu, }, | |
{ exact ⟨1, f.to_hom_units.map_one⟩ } }, | |
-- remaining goal 2 | |
show ∑ b : monoid_hom.range f.to_hom_units, (b : R) = 0, | |
calc ∑ b : monoid_hom.range f.to_hom_units, (b : R) | |
= ∑ n in range (order_of x), x ^ n : | |
eq.symm $ sum_bij (λ n _, x ^ n) | |
(by simp only [mem_univ, forall_true_iff]) | |
(by simp only [implies_true_iff, eq_self_iff_true, subgroup.coe_pow, units.coe_pow, coe_coe]) | |
(λ m n hm hn, pow_injective_of_lt_order_of _ | |
(by simpa only [mem_range] using hm) | |
(by simpa only [mem_range] using hn)) | |
(λ b hb, let ⟨n, hn⟩ := hx b in ⟨n % order_of x, mem_range.2 (nat.mod_lt _ (order_of_pos _)), | |
by rw [← pow_eq_mod_order_of, hn]⟩) | |
... = 0 : _, | |
rw [← mul_left_inj' hx1, zero_mul, geom_sum_mul, coe_coe], | |
norm_cast, | |
simp [pow_order_of_eq_one], | |
end | |
/-- In an integral domain, a sum indexed by a homomorphism from a finite group is zero, | |
unless the homomorphism is trivial, in which case the sum is equal to the cardinality of the group. | |
-/ | |
lemma sum_hom_units (f : G →* R) [decidable (f = 1)] : | |
∑ g : G, f g = if f = 1 then fintype.card G else 0 := | |
begin | |
split_ifs with h h, | |
{ simp [h, card_univ] }, | |
{ exact sum_hom_units_eq_zero f h } | |
end | |
end | |