Theorems · Theorem · commutative algebra

Algebra.norm_eq_prod_embeddings

∀ {L : Type u_6} (K : Type u_7) [inst : Field K] [inst_1 : Field L] [inst_2 : Algebra K L] (E : Type u_9)
  [inst_3 : Field E] [inst_4 : Algebra K E] [inst_5 : FiniteDimensional K L] [Algebra.IsSeparable K L] [IsAlgClosed E]
  (x : L), (algebraMap K E) ((Algebra.norm K) x) = ∏ σ, σ x

For L/K a finite separable extension of fields and E an algebraically closed extension of K, the norm (down to K) of an element x of L is equal to the product of the images of x over all the K-embeddings σ of L into E.

Defined in

Mathlib.RingTheory.Norm.Transitivity

Cited by

5 results in Mathlib

Foundations

Depth 162 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

FieldFieldAlgebraFieldAlgebraFiniteDimensionalAlgebra.IsSeparableIsAlgClosed