Complete Algebra.exists_dvd_nonzero_if_isIntegral (#157)

* create M_spec_map * Algebra.exists_dvd_nonzero_if_isIntegral * Remove `import Mathlib` * Update FLT/MathlibExperiments/FrobeniusRiou.lean Co-authored-by: Pietro Monticone <[email protected]> * Update FLT/MathlibExperiments/FrobeniusRiou.lean Co-authored-by: Pietro Monticone <[email protected]> * Golfing lemmas * Golfing lemmas * cleanup * Rename theorem for refactor --------- Co-authored-by: Kevin Buzzard <[email protected]> Co-authored-by: Pietro Monticone <[email protected]>
ImperialCollegeLondon · Oct 11, 2024 · 04249af · 04249af
1 parent 217555e
commit 04249af
Showing 1 changed file with 31 additions and 5 deletions.
diff --git a/FLT/MathlibExperiments/FrobeniusRiou.lean b/FLT/MathlibExperiments/FrobeniusRiou.lean
@@ -11,7 +11,7 @@ import Mathlib.RingTheory.IntegralClosure.IntegralRestrict
 import Mathlib.RingTheory.Ideal.Pointwise
 import Mathlib.RingTheory.Ideal.Over
 import Mathlib.FieldTheory.Normal
-import Mathlib
+import Mathlib.FieldTheory.SeparableClosure
 import Mathlib.RingTheory.OreLocalization.Ring
 
 /-!
@@ -527,10 +527,36 @@ theorem reduction_isIntegral
 
 end MulSemiringAction
 
-theorem Algebra.exists_dvd_nonzero_if_isIntegral (R S : Type) [CommRing R] [CommRing S]
- [Algebra R S] [Algebra.IsIntegral R S] [IsDomain S] (s : S) (hs : s ≠ 0) :
- ∃ r : R, r ≠ 0 ∧ s ∣ (r : S) := by
- sorry
+theorem Polynomial.nonzero_const_if_isIntegral (R S : Type) [CommRing R] [Nontrivial R]
+ [CommRing S] [Algebra R S] [Algebra.IsIntegral R S] [IsDomain S] (s : S) (hs : s ≠ 0) :
+ ∃ (q : Polynomial R), q.coeff 0 ≠ 0 ∧ q.eval₂ (algebraMap R S) s = 0 := by
+ obtain ⟨p, p_monic, p_eval⟩ := (@Algebra.isIntegral_def R S).mp inferInstance s
+ have p_nzero := ne_zero_of_ne_zero_of_monic X_ne_zero p_monic
+ obtain ⟨q, p_eq, q_ndvd⟩ := Polynomial.exists_eq_pow_rootMultiplicity_mul_and_not_dvd p p_nzero 0
+ rw [C_0, sub_zero] at p_eq
+ rw [C_0, sub_zero] at q_ndvd
+ use q
+ constructor
+ . intro q_coeff_0
+ exact q_ndvd <| X_dvd_iff.mpr q_coeff_0
+ . rw [p_eq, eval₂_mul] at p_eval
+ rcases NoZeroDivisors.eq_zero_or_eq_zero_of_mul_eq_zero p_eval with Xpow_eval | q_eval
+ . by_contra
+ apply hs
+ rw [eval₂_X_pow] at Xpow_eval
+ exact pow_eq_zero Xpow_eval
+ . exact q_eval
+
+theorem Algebra.exists_dvd_nonzero_if_isIntegral (R S : Type) [CommRing R] [Nontrivial R]
+ [CommRing S] [Algebra R S] [Algebra.IsIntegral R S] [IsDomain S] (s : S) (hs : s ≠ 0) :
+ ∃ r : R, r ≠ 0 ∧ s ∣ (algebraMap R S) r := by
+ obtain ⟨q, q_zero_coeff, q_eval_zero⟩ := Polynomial.nonzero_const_if_isIntegral R S s hs
+ use q.coeff 0
+ refine ⟨q_zero_coeff, ?_⟩
+ rw [← Polynomial.eval₂_X (algebraMap R S) s, ← dvd_neg, ← Polynomial.eval₂_C (algebraMap R S) s]
+ rw [← zero_add (-_), Mathlib.Tactic.RingNF.add_neg, ← q_eval_zero, ← Polynomial.eval₂_sub]
+ apply Polynomial.eval₂_dvd
+ exact Polynomial.X_dvd_sub_C
 
 end B_mod_Q_over_A_mod_P_stuff