mathlib documentation

number_theory.kummer_dedekind

Kummer-Dedekind theorem #

This file proves the monogenic version of the Kummer-Dedekind theorem on the splitting of prime ideals in an extension of the ring of integers. This states that if I is a prime ideal of Dedekind domain R and S = R[α] for some α that is integral over R with minimal polynomial f, then the prime factorisations of I * S and f mod I have the same shape, i.e. they have the same number of prime factors, and each prime factors of I * S can be paired with a prime factor of f mod I in a way that ensures multiplicities match (in fact, this pairing can be made explicit with a formula).

Main definitions #

Main results #

TODO #

References #

Tags #

kummer, dedekind, kummer dedekind, dedekind-kummer, dedekind kummer

def conductor (R : Type u_1) {S : Type u_2} [comm_ring R] [comm_ring S] [algebra R S] (x : S) :

Let S / R be a ring extension and x : S, then the conductor of R<x> is the biggest ideal of S contained in R<x>.

Equations
theorem conductor_eq_of_eq {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] [algebra R S] {x y : S} (h : (algebra.adjoin R {x}) = (algebra.adjoin R {y})) :
theorem conductor_subset_adjoin {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] [algebra R S] {x : S} :
theorem mem_conductor_iff {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] [algebra R S] {x y : S} :
y conductor R x (b : S), y * b algebra.adjoin R {x}
theorem prod_mem_ideal_map_of_mem_conductor {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] [algebra R S] {x : S} {I : ideal R} {p : R} {z : S} (hp : p ideal.comap (algebra_map R S) (conductor R x)) (hz' : z ideal.map (algebra_map R S) I) :

This technical lemma tell us that if C is the conductor of R<x> and I is an ideal of R then p * (I * S) ⊆ I * R<x> for any p in C ∩ R

A technical result telling us that (I * S) ∩ R<x> = I * R<x> for any ideal I of R.

noncomputable def quot_adjoin_equiv_quot_map {R : Type u_1} {S : Type u_2} [comm_ring R] [comm_ring S] [algebra R S] {x : S} {I : ideal R} (hx : ideal.comap (algebra_map R S) (conductor R x) I = ) (h_alg : function.injective (algebra_map (algebra.adjoin R {x}) S)) :

The canonical morphism of rings from R<x> ⧸ (I*R<x>) to S ⧸ (I*S) is an isomorphism when I and (conductor R x) ∩ R are coprime.

Equations

The first half of the Kummer-Dedekind Theorem in the monogenic case, stating that the prime factors of I*S are in bijection with those of the minimal polynomial of the generator of S over R, taken mod I.

Equations

The second half of the Kummer-Dedekind Theorem in the monogenic case, stating that the bijection factors_equiv' defined in the first half preserves multiplicities.