Theorems · Theorem · number theory
Polynomial.X_pow_sub_one_splits
∀ {K : Type u_1} [inst : Field K] {ζ : K} {n : ℕ}, IsPrimitiveRoot ζ n → (Polynomial.X ^ n - Polynomial.C 1).SplitsIf there is a primitive n-th root of unity in K, then X ^ n - 1 splits.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 140 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Field
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites14
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
- RingHomstatement · cited by 10,189
- Fieldstatement and proof · cited by 7,404
- Polynomialstatement · cited by 5,681
- Multisetproof · cited by 2,627
- Polynomial.Xstatement and proof · cited by 1,639
- Polynomial.Cstatement and proof · cited by 1,598
- Polynomial.natDegreeproof · cited by 1,105
- Multiset.cardproof · cited by 375
- IsPrimitiveRootstatement and proof · cited by 356
- Polynomial.Splitsstatement · cited by 290
- Polynomial.splits_iff_card_rootsproof · cited by 11
Cited by1
Results whose statement or proof uses this declaration.
- IsCyclotomicExtension.splits_X_pow_sub_oneproof · cited by 2