Theorems · Theorem · number theory

Ideal.exists_smul_eq_of_isGaloisGroup

∀ {A : Type u_1} {B : Type u_2} [inst : CommRing A] [inst_1 : CommRing B] [inst_2 : Algebra A B] (p : Ideal A)
  (P Q : Ideal B) [hPp : P.IsPrime] [hp : P.LiesOver p] [hQp : Q.IsPrime] [Q.LiesOver p] (G : Type u_3)
  [inst_4 : Group G] [Finite G] [inst_6 : MulSemiringAction G B] [IsGaloisGroup G A B], ∃ σ, σ • P = Q

If p is a maximal ideal of A, P and Q are prime ideals lying over p, then there exists σ ∈ Aut (B / A) such that σ P = Q. In other words, the Galois group Gal(L / K) acts transitively on the set of all prime ideals lying over p.

Defined in

Mathlib.NumberTheory.RamificationInertia.Galois

Cited by

6 results in Mathlib

Foundations

Depth 99 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraIdeal.IsPrimeIdeal.LiesOverIdeal.IsPrimeIdeal.LiesOverGroupFiniteMulSemiringActionIsGaloisGroup