map_cyclotomic_int — Mathlib

English

The map that sends integers into R preserves cyclotomic polynomials: map from ℤ to R of cyclotomic n equals cyclotomic n in R.

Русский

Отображение целочисленного polynom в кольцо R сохраняет циклотоми́ческие: map(ℤ → R) cyclotomic n ℤ = cyclotomic n R.

LaTeX

$$$\operatorname{map}(\operatorname{Int.castRingHom}R)\big( \operatorname{cyclotomic}(n,\mathbb{Z}) \big) = \operatorname{cyclotomic}(n,R)$$$

Lean4

/-- `cyclotomic n R` comes from `cyclotomic n ℤ`. -/
theorem map_cyclotomic_int (n : ℕ) (R : Type*) [Ring R] : map (Int.castRingHom R) (cyclotomic n ℤ) = cyclotomic n R :=
  by
  by_cases hzero : n = 0
  · simp only [hzero, cyclotomic, dif_pos, Polynomial.map_one]
  simp [cyclotomic, hzero]

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