English
If p and q are coprime polynomials in R[X], then the kernels of aeval f applied to p and q are disjoint as submodules of M.
Русский
Если p и q COPRIME полиномы в R[X], ядра aeval f(p) и aeval f(q) являются взаимно-дисjoint.
LaTeX
$$Disjoint (ker (aeval f p)) (ker (aeval f q))$$
Lean4
/-- **Hilbert basis theorem**: a polynomial ring over a Noetherian ring is a Noetherian ring. -/
protected theorem isNoetherianRing [inst : IsNoetherianRing R] : IsNoetherianRing R[X] :=
isNoetherianRing_iff.2
⟨fun I : Ideal R[X] =>
let M := inst.wf.min (Set.range I.leadingCoeffNth) ⟨_, ⟨0, rfl⟩⟩
have hm : M ∈ Set.range I.leadingCoeffNth := WellFounded.min_mem _ _ _
let ⟨N, HN⟩ := hm
let ⟨s, hs⟩ := I.is_fg_degreeLE N
have hm2 : ∀ k, I.leadingCoeffNth k ≤ M := fun k =>
Or.casesOn (le_or_gt k N) (fun h => HN ▸ I.leadingCoeffNth_mono h) fun h _ hx =>
Classical.by_contradiction fun hxm =>
haveI : IsNoetherian R R := inst
have : ¬M < I.leadingCoeffNth k := by refine WellFounded.not_lt_min inst.wf _ _ ?_; exact ⟨k, rfl⟩
this ⟨HN ▸ I.leadingCoeffNth_mono (le_of_lt h), fun H => hxm (H hx)⟩
have hs2 : ∀ {x}, x ∈ I.degreeLE N → x ∈ Ideal.span (↑s : Set R[X]) :=
hs ▸ fun hx =>
Submodule.span_induction (hx := hx) (fun _ hx => Ideal.subset_span hx) (Ideal.zero_mem _)
(fun _ _ _ _ => Ideal.add_mem _) fun c f _ hf => f.C_mul' c ▸ Ideal.mul_mem_left _ _ hf
⟨s,
le_antisymm
(Ideal.span_le.2 fun x hx =>
have : x ∈ I.degreeLE N := hs ▸ Submodule.subset_span hx
this.2) <|
by
have : Submodule.span R[X] ↑s = Ideal.span ↑s := rfl
rw [this]
intro p hp
generalize hn : p.natDegree = k
induction k using Nat.strong_induction_on generalizing p with
| _ k ih
rcases le_or_gt k N with h | h
· subst k
refine
hs2 ⟨Polynomial.mem_degreeLE.2 (le_trans Polynomial.degree_le_natDegree <| WithBot.coe_le_coe.2 h), hp⟩
· have hp0 : p ≠ 0 := by
rintro rfl
cases hn
exact Nat.not_lt_zero _ h
have : (0 : R) ≠ 1 := by
intro h
apply hp0
ext i
refine (mul_one _).symm.trans ?_
rw [← h, mul_zero]
rfl
haveI : Nontrivial R := ⟨⟨0, 1, this⟩⟩
have : p.leadingCoeff ∈ I.leadingCoeffNth N := by
rw [HN]
exact hm2 k ((I.mem_leadingCoeffNth _ _).2 ⟨_, hp, hn ▸ Polynomial.degree_le_natDegree, rfl⟩)
rw [I.mem_leadingCoeffNth] at this
rcases this with ⟨q, hq, hdq, hlqp⟩
have hq0 : q ≠ 0 := by
intro H
rw [← Polynomial.leadingCoeff_eq_zero] at H
rw [hlqp, Polynomial.leadingCoeff_eq_zero] at H
exact hp0 H
have h1 : p.degree = (q * Polynomial.X ^ (k - q.natDegree)).degree :=
by
rw [Polynomial.degree_mul', Polynomial.degree_X_pow]
· rw [Polynomial.degree_eq_natDegree hp0, Polynomial.degree_eq_natDegree hq0]
rw [← Nat.cast_add, add_tsub_cancel_of_le, hn]
· refine le_trans (Polynomial.natDegree_le_of_degree_le hdq) (le_of_lt h)
rw [Polynomial.leadingCoeff_X_pow, mul_one]
exact mt Polynomial.leadingCoeff_eq_zero.1 hq0
have h2 : p.leadingCoeff = (q * Polynomial.X ^ (k - q.natDegree)).leadingCoeff := by
rw [← hlqp, Polynomial.leadingCoeff_mul_X_pow]
have := Polynomial.degree_sub_lt h1 hp0 h2
rw [Polynomial.degree_eq_natDegree hp0] at this
rw [← sub_add_cancel p (q * Polynomial.X ^ (k - q.natDegree))]
convert (Ideal.span ↑s).add_mem _ ((Ideal.span (s : Set R[X])).mul_mem_right _ _)
· by_cases hpq : p - q * Polynomial.X ^ (k - q.natDegree) = 0
· rw [hpq]
exact Ideal.zero_mem _
refine ih _ ?_ (I.sub_mem hp (I.mul_mem_right _ hq)) rfl
rwa [Polynomial.degree_eq_natDegree hpq, Nat.cast_lt, hn] at this
exact hs2 ⟨Polynomial.mem_degreeLE.2 hdq, hq⟩⟩⟩
