Theorems · Theorem · commutative algebra

Ideal.exists_relNorm_eq_pow_of_isPrime

∀ {R : Type u_1} [inst : CommRing R] [inst_1 : IsDomain R] {S : Type u_3} [inst_2 : CommRing S] [inst_3 : IsDomain S]
  [inst_4 : IsIntegrallyClosed R] [inst_5 : IsIntegrallyClosed S] [inst_6 : Algebra R S] [inst_7 : Module.Finite R S]
  [inst_8 : Module.IsTorsionFree R S] [inst_9 : IsDedekindDomain R] [inst_10 : IsDedekindDomain S] (P : Ideal S)
  (p : Ideal R) [hPp : P.LiesOver p] [p.IsPrime], ∃ s, (Ideal.relNorm R) P = p ^ s

See Ideal.relNorm_eq_pow_of_isMaximal for a more precise statement when p is a maximal ideal.

Defined in: Mathlib.RingTheory.Ideal.Norm.RelNorm
Cited by: 1 results in Mathlib
Foundations: Depth 193 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing IsDomain CommRing IsDomain IsIntegrallyClosed IsIntegrallyClosed Algebra Module.Finite Module.IsTorsionFree IsDedekindDomain IsDedekindDomain Ideal.LiesOver Ideal.IsPrime

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CommRingstatement and proof · cited by 17,173
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one_ne_zeroproof · cited by 885
Ideal.IsPrimestatement and proof · cited by 827
MonoidWithZeroHomstatement · cited by 704

Cited by1

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Ideal.relNorm_eq_pow_of_isPrime_isGaloisproof · cited by 1