feat(number_theory): Dedekind-Kummer theorem (#15000)

It's PR #15000! The Kummer-Dedekind theorem on the splitting of ideals in monogenic ring extensions.

[Zulip thread](https://leanprover.zulipchat.com/#narrow/stream/144837-PR-reviews/topic/.2315000.3A.20the.20Kummer-Dedekind.20theorem)

Co-Authored-By: Paul Lezeau <paul.lezeau@gmail.com>



Co-authored-by: Paul-Lez <72892199+Paul-Lez@users.noreply.github.com>
Co-authored-by: Johan Commelin <johan@commelin.net>
Co-authored-by: Anne Baanen <t.baanen@vu.nl>

Author: 3 people

src/data/multiset/basic.lean

@@ -1104,6 +1104,9 @@ lemma attach_cons (a : α) (m : multiset α) :
 quotient.induction_on m $ assume l, congr_arg coe $ congr_arg (list.cons _) $
   by rw [list.map_pmap]; exact list.pmap_congr _ (λ _ _ _ _, subtype.eq rfl)
 
+@[simp]
+lemma attach_map_coe (m : multiset α) : multiset.map (coe : _ → α) m.attach = m := m.attach_map_val
+
 section decidable_pi_exists
 variables {m : multiset α}
 
@@ -1866,6 +1869,13 @@ begin
     contradiction }
 end
 
+@[simp]
+lemma attach_count_eq_count_coe (m : multiset α) (a) : m.attach.count a = m.count (a : α) :=
+calc m.attach.count a
+    = (m.attach.map (coe : _ → α)).count (a : α) :
+  (multiset.count_map_eq_count' _ _ subtype.coe_injective _).symm
+... = m.count (a : α) : congr_arg _ m.attach_map_coe
+
 lemma filter_eq' (s : multiset α) (b : α) : s.filter (= b) = repeat b (count b s) :=
 begin
   ext a,

src/number_theory/kummer_dedekind.lean

@@ -0,0 +1,168 @@
+/-
+Copyright (c) 2021 Anne Baanen. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Anne Baanen, Paul Lezeau
+-/
+
+import ring_theory.adjoin_root
+import ring_theory.dedekind_domain.ideal
+import ring_theory.algebra_tower
+
+/-!
+# Kummer-Dedekind theorem
+
+This file proves the monogenic version of the Kummer-Dedekind theorem on the splitting of prime
+ideals in an extension of the ring of integers. This states that if `I` is a prime ideal of
+Dedekind domain `R` and `S = R[α]` for some `α` that is integral over `R` with minimal polynomial
+`f`, then the prime factorisations of `I * S` and `f mod I` have the same shape, i.e. they have the
+same number of prime factors, and each prime factors of `I * S` can be paired with a prime factor
+of `f mod I` in a way that ensures multiplicities match (in fact, this pairing can be made explicit
+with a formula).
+
+## Main definitions
+
+ * `normalized_factors_map_equiv_normalized_factors_min_poly_mk` : The bijection in the
+    Kummer-Dedekind theorem. This is the pairing between the prime factors of `I * S` and the prime
+    factors of `f mod I`.
+
+## Main results
+
+ * `normalized_factors_ideal_map_eq_normalized_factors_min_poly_mk_map` : The Kummer-Dedekind
+    theorem.
+ * `ideal.irreducible_map_of_irreducible_minpoly` : `I.map (algebra_map R S)` is irreducible if
+    `(map I^.quotient.mk (minpoly R pb.gen))` is irreducible, where `pb` is a power basis of `S`
+    over `R`.
+
+## TODO
+
+ * Define the conductor ideal and prove the Kummer-Dedekind theorem in full generality.
+
+ * Prove the converse of `ideal.irreducible_map_of_irreducible_minpoly`.
+
+ * Prove that `normalized_factors_map_equiv_normalized_factors_min_poly_mk` can be expressed as
+    `normalized_factors_map_equiv_normalized_factors_min_poly_mk g = ⟨I, G(α)⟩` for `g` a prime
+    factor of `f mod I` and `G` a lift of `g` to `R[X]`.
+
+## References
+
+ * [J. Neukirch, *Algebraic Number Theory*][Neukirch1992]
+
+## Tags
+
+kummer, dedekind, kummer dedekind, dedekind-kummer, dedekind kummer
+-/
+
+namespace kummer_dedekind
+
+open_locale big_operators polynomial classical
+
+open ideal polynomial double_quot unique_factorization_monoid
+
+variables {R : Type*} [comm_ring R]
+variables {S : Type*} [comm_ring S] [is_domain S] [is_dedekind_domain S] [algebra R S]
+variables (pb : power_basis R S) {I : ideal R}
+
+local attribute [instance] ideal.quotient.field
+
+variables [is_domain R]
+
+/-- The first half of the **Kummer-Dedekind Theorem** in the monogenic case, stating that the prime
+    factors of `I*S` are in bijection with those of the minimal polynomial of the generator of `S`
+    over `R`, taken `mod I`.-/
+noncomputable def normalized_factors_map_equiv_normalized_factors_min_poly_mk (hI : is_maximal I)
+  (hI' : I ≠ ⊥) : {J : ideal S | J ∈ normalized_factors (I.map (algebra_map R S) )} ≃
+    {d : (R ⧸ I)[X] | d ∈ normalized_factors (map I^.quotient.mk (minpoly R pb.gen)) } :=
+((normalized_factors_equiv_of_quot_equiv ↑(pb.quotient_equiv_quotient_minpoly_map I)
+  --show that `I * S` ≠ ⊥
+  (show I.map (algebra_map R S) ≠ ⊥,
+    by rwa [ne.def, map_eq_bot_iff_of_injective pb.basis.algebra_map_injective, ← ne.def])
+  --show that the ideal spanned by `(minpoly R pb.gen) mod I` is non-zero
+  (by {by_contra, exact (show (map I^.quotient.mk (minpoly R pb.gen) ≠ 0), from
+    polynomial.map_monic_ne_zero (minpoly.monic pb.is_integral_gen))
+    (span_singleton_eq_bot.mp h) } )).trans
+  (normalized_factors_equiv_span_normalized_factors
+    (show (map I^.quotient.mk (minpoly R pb.gen)) ≠ 0, from
+      polynomial.map_monic_ne_zero (minpoly.monic pb.is_integral_gen))).symm)
+
+/-- The second half of the **Kummer-Dedekind Theorem** in the monogenic case, stating that the
+    bijection `factors_equiv'` defined in the first half preserves multiplicities. -/
+theorem multiplicity_factors_map_eq_multiplicity (hI : is_maximal I) (hI' : I ≠ ⊥) {J : ideal S}
+  (hJ : J ∈ normalized_factors (I.map (algebra_map R S))) :
+  multiplicity J (I.map (algebra_map R S)) =
+    multiplicity ↑(normalized_factors_map_equiv_normalized_factors_min_poly_mk pb hI hI' ⟨J, hJ⟩)
+    (map I^.quotient.mk (minpoly R pb.gen)) :=
+by rw [normalized_factors_map_equiv_normalized_factors_min_poly_mk, equiv.coe_trans,
+       function.comp_app,
+       multiplicity_normalized_factors_equiv_span_normalized_factors_symm_eq_multiplicity,
+       normalized_factors_equiv_of_quot_equiv_multiplicity_eq_multiplicity]
+
+/-- The **Kummer-Dedekind Theorem**. -/
+theorem normalized_factors_ideal_map_eq_normalized_factors_min_poly_mk_map (hI : is_maximal I)
+  (hI' : I ≠ ⊥) : normalized_factors (I.map (algebra_map R S)) = multiset.map
+  (λ f, ((normalized_factors_map_equiv_normalized_factors_min_poly_mk pb hI hI').symm f : ideal S))
+      (normalized_factors (polynomial.map I^.quotient.mk (minpoly R pb.gen))).attach :=
+begin
+  ext J,
+  -- WLOG, assume J is a normalized factor
+  by_cases hJ : J ∈ normalized_factors (I.map (algebra_map R S)), swap,
+  { rw [multiset.count_eq_zero.mpr hJ, eq_comm, multiset.count_eq_zero, multiset.mem_map],
+    simp only [multiset.mem_attach, true_and, not_exists],
+    rintros J' rfl,
+    exact hJ
+      ((normalized_factors_map_equiv_normalized_factors_min_poly_mk pb hI hI').symm J').prop },
+
+  -- Then we just have to compare the multiplicities, which we already proved are equal.
+  have := multiplicity_factors_map_eq_multiplicity pb hI hI' hJ,
+  rw [multiplicity_eq_count_normalized_factors, multiplicity_eq_count_normalized_factors,
+      unique_factorization_monoid.normalize_normalized_factor _ hJ,
+      unique_factorization_monoid.normalize_normalized_factor,
+      part_enat.coe_inj]
+    at this,
+  refine this.trans _,
+  -- Get rid of the `map` by applying the equiv to both sides.
+  generalize hJ' : (normalized_factors_map_equiv_normalized_factors_min_poly_mk pb hI hI')
+    ⟨J, hJ⟩ = J',
+  have : ((normalized_factors_map_equiv_normalized_factors_min_poly_mk pb hI hI').symm
+    J' : ideal S) = J,
+  { rw [← hJ', equiv.symm_apply_apply _ _, subtype.coe_mk] },
+  subst this,
+  -- Get rid of the `attach` by applying the subtype `coe` to both sides.
+  rw [multiset.count_map_eq_count' (λ f,
+      ((normalized_factors_map_equiv_normalized_factors_min_poly_mk pb hI hI').symm f
+        : ideal S)),
+      multiset.attach_count_eq_count_coe],
+  { exact subtype.coe_injective.comp (equiv.injective _) },
+  { exact (normalized_factors_map_equiv_normalized_factors_min_poly_mk pb hI hI' _).prop },
+  { exact irreducible_of_normalized_factor _
+    (normalized_factors_map_equiv_normalized_factors_min_poly_mk pb hI hI' _).prop },
+  { exact polynomial.map_monic_ne_zero (minpoly.monic pb.is_integral_gen) },
+  { exact irreducible_of_normalized_factor _ hJ },
+  { rwa [← bot_eq_zero, ne.def, map_eq_bot_iff_of_injective pb.basis.algebra_map_injective] },
+end
+
+theorem ideal.irreducible_map_of_irreducible_minpoly (hI : is_maximal I) (hI' : I ≠ ⊥)
+  (hf : irreducible (map I^.quotient.mk (minpoly R pb.gen))) :
+  irreducible (I.map (algebra_map R S)) :=
+begin
+  have mem_norm_factors : normalize (map I^.quotient.mk (minpoly R pb.gen)) ∈ normalized_factors
+    (map I^.quotient.mk (minpoly R pb.gen)) := by simp [normalized_factors_irreducible hf],
+  suffices : ∃ x, normalized_factors (I.map (algebra_map R S)) = {x},
+  { obtain ⟨x, hx⟩ := this,
+    have h := normalized_factors_prod (show I.map (algebra_map R S) ≠ 0, by
+      rwa [← bot_eq_zero, ne.def, map_eq_bot_iff_of_injective pb.basis.algebra_map_injective]),
+    rw [associated_iff_eq, hx, multiset.prod_singleton] at h,
+    rw ← h,
+    exact irreducible_of_normalized_factor x
+      (show x ∈ normalized_factors (I.map (algebra_map R S)), by simp [hx]) },
+  rw normalized_factors_ideal_map_eq_normalized_factors_min_poly_mk_map pb hI hI',
+  use ((normalized_factors_map_equiv_normalized_factors_min_poly_mk pb hI hI').symm
+    ⟨normalize (map I^.quotient.mk (minpoly R pb.gen)), mem_norm_factors⟩ : ideal S),
+  rw multiset.map_eq_singleton,
+  use ⟨normalize (map I^.quotient.mk (minpoly R pb.gen)), mem_norm_factors⟩,
+  refine ⟨_, rfl⟩,
+  apply multiset.map_injective subtype.coe_injective,
+  rw [multiset.attach_map_coe, multiset.map_singleton, subtype.coe_mk],
+  exact normalized_factors_irreducible hf
+end
+
+end kummer_dedekind

src/ring_theory/ideal/operations.lean

@@ -1530,6 +1530,7 @@ by { rw [set_like.ext'_iff, ker_eq, set.ext_iff], exact injective_iff_map_eq_zer
 
 lemma ker_eq_bot_iff_eq_zero : ker f = ⊥ ↔ ∀ x, f x = 0 → x = 0 :=
 by { rw [← injective_iff_map_eq_zero f, injective_iff_ker_eq_bot] }
+
 omit rc
 
 @[simp] lemma ker_coe_equiv (f : R ≃+* S) :
@@ -1677,6 +1678,11 @@ begin
       abel },
     exact (H.mem_or_mem this).imp (λ h, ha ▸ mem_map_of_mem f h) (λ h, hb ▸ mem_map_of_mem f h) }
 end
+
+lemma map_eq_bot_iff_of_injective {I : ideal R} {f : F} (hf : function.injective f) :
+  I.map f = ⊥ ↔ I = ⊥ :=
+by rw [map_eq_bot_iff_le_ker, (ring_hom.injective_iff_ker_eq_bot f).mp hf, le_bot_iff]
+
 omit rc
 
 theorem map_is_prime_of_equiv {F' : Type*} [ring_equiv_class F' R S]