Theorems · Theorem · number theory
IsPrimitiveRoot.toInteger_sub_one_dvd_prime
∀ {p k : ℕ} {K : Type u} [inst : Field K] {ζ : K} [hp : Fact (Nat.Prime p)] [inst_1 : CharZero K]
[hcycl : IsCyclotomicExtension {p ^ (k + 1)} ℚ K] (hζ : IsPrimitiveRoot ζ (p ^ (k + 1))), hζ.toInteger - 1 ∣ ↑pIn a p ^ (k + 1)-th cyclotomic extension of ℚ, we have that
ζ - 1 divides p in 𝓞 K.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 219 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites26
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
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- Setstatement · cited by 53,352
- Fieldstatement and proof · cited by 7,404
- Factstatement and proof · cited by 2,726
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- zero_addproof · cited by 2,366
- Nat.Primestatement and proof · cited by 2,059
- pow_zeroproof · cited by 1,094
- CharZerostatement and proof · cited by 932
- pow_oneproof · cited by 894
- NumberFieldproof · cited by 653
- NumberField.RingOfIntegersstatement and proof · cited by 413
Cited by1
Results whose statement or proof uses this declaration.
- IsPrimitiveRoot.toInteger_sub_one_dvd_prime'proof · cited by 1