Theorems · Theorem · number theory
IsCyclotomicExtension.Rat.Three.cube_sub_one_eq_mul
∀ {K : Type u_1} [inst : Field K] {ζ : K} (hζ : IsPrimitiveRoot ζ 3) (x : NumberField.RingOfIntegers K),
x ^ 3 - 1 = (x - 1) * (x - ↑⋯.unit) * (x - ↑⋯.unit ^ 2)We have that x ^ 3 - 1 = (x - 1) * (x - η) * (x - η ^ 2).
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 209 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- Field
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Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Fieldstatement and proof · cited by 7,404
- Nat.cast_oneproof · cited by 2,501
- Units.valstatement and proof · cited by 1,966
- Nat.cast_zeroproof · cited by 1,870
- NumberField.RingOfIntegersstatement and proof · cited by 413
- IsPrimitiveRootstatement and proof · cited by 356
- IsUnit.unitstatement and proof · cited by 252
- IsPrimitiveRoot.toIntegerstatement and proof · cited by 72
- IsPrimitiveRoot.isUnitstatement and proof · cited by 22
- IsPrimitiveRoot.toInteger_isPrimitiveRootstatement and proof · cited by 15
- IsPrimitiveRoot.toInteger_cube_eq_oneproof · cited by 1
- IsCyclotomicExtension.Rat.Three.eta_sq_add_eta_add_oneproof · cited by 1
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