English
Similar to the previous result: trace is preserved under suitably compatible equivalences of the coefficient rings and the target rings.
Русский
Аналогично предыдущему результату: след сохраняется при совместимо-совместимых эквивалентностях коэффициентных колец и целевых колец.
LaTeX
$$Same as 185525 in formal form: $\\operatorname{trace}$ invariance under equivs$$
Lean4
theorem trace_eq_sum_embeddings_gen (pb : PowerBasis K L) (hE : (minpoly K pb.gen).Splits (algebraMap K E))
(hfx : IsSeparable K pb.gen) :
algebraMap K E (Algebra.trace K L pb.gen) = (@Finset.univ _ (PowerBasis.AlgHom.fintype pb)).sum fun σ => σ pb.gen :=
by
letI := Classical.decEq E
letI : Fintype (L →ₐ[K] E) := PowerBasis.AlgHom.fintype pb
rw [pb.trace_gen_eq_sum_roots hE, Fintype.sum_equiv pb.liftEquiv', Finset.sum_mem_multiset,
Finset.sum_eq_multiset_sum, Multiset.toFinset_val, Multiset.dedup_eq_self.mpr _, Multiset.map_id]
· exact nodup_roots ((separable_map _).mpr hfx)
swap
· intro x; rfl
· intro σ
rw [PowerBasis.liftEquiv'_apply_coe, id_def]
