English
If S is a set of natural numbers and F1, F2 are two intermediate fields with IsCyclotomicExtension S K (F1) and IsCyclotomicExtension S K (F2), then F1 = F2.
Русский
Если множество S содержит степени цикла и F1, F2 — два промежуточных поля с IsCyclotomicExtension S K (F1) и IsCyclotomicExtension S K (F2), то F1 = F2.
LaTeX
$$$\\text{If }F_1,F_2\\text{ satisfy IsCyclotomicExtension}(S,K,F_i),\\; i=1,2,\\; \\text{then }F_1=F_2$$$
Lean4
theorem le_of_dvd [NeZero n₂] (h : n₁ ∣ n₂) : C₁ ≤ C₂ :=
by
by_cases hn₁ : n₁ = 0
· rw [hn₁, zero_dvd_iff] at h
exact False.elim <| NeZero.ne n₂ h
have : NeZero n₁ := ⟨hn₁⟩
obtain ⟨ζ₂, hζ₂⟩ := h₂.1 rfl (NeZero.ne n₂)
replace hζ₂ := hζ₂.map_of_injective (FaithfulSMul.algebraMap_injective C₂ B)
obtain ⟨d, hd⟩ := h
have hζ₁ := IsPrimitiveRoot.pow n₂.pos_of_neZero hζ₂ (by rwa [mul_comm])
simpa [(isCyclotomicExtension_singleton_iff_eq_adjoin n₁ C₁ hζ₁).mp h₁,
(isCyclotomicExtension_singleton_iff_eq_adjoin n₂ C₂ hζ₂).mp h₂] using
adjoin_le <| Set.singleton_subset_iff.mpr <| Subalgebra.pow_mem _ (self_mem_adjoin_singleton A _) _
