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]