Theorems · Theorem · field theory

X_pow_sub_C_eq_prod

∀ {R : Type u_1} [inst : CommRing R] [IsDomain R] {n : ℕ} {ζ : R},
  IsPrimitiveRoot ζ n →
    ∀ {α a : R},
      0 < n →
        α ^ n = a → Polynomial.X ^ n - Polynomial.C a = ∏ i ∈ Finset.range n, (Polynomial.X - Polynomial.C (ζ ^ i * α))

Defined in: Mathlib.FieldTheory.KummerExtension
Cited by: 2 results in Mathlib
Foundations: Depth 142 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing IsDomain

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Cites25

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

DFunLike.coestatement and proof · cited by 62,936
CommRingstatement and proof · cited by 17,173
RingHomstatement and proof · cited by 10,189
Polynomialstatement · cited by 5,681
Algebra.algebraMapproof · cited by 4,706
Finset.prodstatement and proof · cited by 2,356
IsDomainstatement and proof · cited by 2,196
Polynomial.Xstatement and proof · cited by 1,639
Polynomial.Cstatement and proof · cited by 1,598
Finset.rangestatement and proof · cited by 1,341
map_mulproof · cited by 1,137
Polynomial.mapproof · cited by 806

Cited by2

Results whose statement or proof uses this declaration.

IsPrimitiveRoot.prod_one_sub_pow_eq_orderproof · cited by 1
Polynomial.separable_X_pow_sub_C_of_irreducibleproof · cited by 1