Theorems · Theorem · commutative algebra

associated_norm_prod_smith

∀ {R : Type u_1} {S : Type u_2} {ι : Type u_3} [inst : CommRing R] [inst_1 : IsDomain R]
  [inst_2 : IsPrincipalIdealRing R] [inst_3 : CommRing S] [inst_4 : IsDomain S] [inst_5 : Algebra R S]
  [inst_6 : Fintype ι] (b : Module.Basis ι R S) {f : S} (hf : f ≠ 0),
  Associated ((Algebra.norm R) f) (∏ i, Ideal.smithCoeffs b (Ideal.span {f}) ⋯ i)

For a nonzero element f in an algebra S over a principal ideal domain R that is finite and free as an R-module, the norm of f relative to R is associated to the product of the Smith coefficients of the ideal generated by f.

Defined in

Mathlib.LinearAlgebra.FreeModule.Norm

Cited by

1 results in Mathlib

Foundations

Depth 131 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingIsDomainIsPrincipalIdealRingCommRingIsDomainAlgebraFintype