English
Under the same Kummer–Dedekind hypotheses, if the minimal polynomial of x modulo I is irreducible after mapping, then the image ideal I.map(algebraMap R S) is irreducible.
Русский
При тех же условиях Куммер-Дедек, если образ минимального полинома x по модулю I неприводим, то образ I в S через алгебраическую карту неприводим.
LaTeX
$$
(hI : IsMaximal) (hI' : I ≠ ⊥) (hx : (conductor R x).comap (algebraMap R S) ⊔ I = ⊤) (hx' : IsIntegral R x) (hf : Irreducible (Polynomial.map (Ideal.Quotient.mk I) (minpoly R x))) : Irreducible (Ideal.map (algebraMap R S) I)$$
Lean4
/-- Reformulated functional equation with `f` and `g` interchanged. -/
theorem h_feq' (P : WeakFEPair E) (x : ℝ) (hx : 0 < x) : P.g (1 / x) = (P.ε⁻¹ * ↑(x ^ P.k)) • P.f x :=
by
rw [(div_div_cancel₀ (one_ne_zero' ℝ) ▸ P.h_feq (1 / x) (one_div_pos.mpr hx) :), ← mul_smul]
convert (one_smul ℂ (P.g (1 / x))).symm using 2
rw [one_div, inv_rpow hx.le, ofReal_inv]
field_simp [P.hε, (rpow_pos_of_pos hx _).ne']
