natDegree_eq_toNat_order_map_of_ne_top

English

For a Weierstrass factorization with I ≠ ⊤, the degree of f equals the lifted order of g modulo I.

Русский

При факторизации Вайершстраса с I ≠ ⊤, deg f равен поднесённому порядку образа g по I.

LaTeX

$$f.degree = (g.map (Ideal.Quotient.mk I)).order.lift (order_finite_iff_ne_zero.2 (H.map_ne_zero_of_ne_top hI))$$

Lean4

theorem natDegree_eq_toNat_order_map_of_ne_top (hI : I ≠ ⊤) : f.natDegree = (g.map (Ideal.Quotient.mk I)).order.toNat :=
  by
  rw [Polynomial.natDegree, H.degree_eq_coe_lift_order_map_of_ne_top hI, ENat.lift_eq_toNat_of_lt_top]
  exact WithBot.unbotD_coe _ _

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