trace_surjective — Mathlib

English

For a separable finite extension K ⊂ L and a basis b of L over K, the determinant of the trace form matrix is nonzero.

Русский

Для разделяемого конечного расширения K ⊂ L и базиса b базиса L над KDet равен не нулю.

LaTeX

$$$\\det\\big(\\mathrm{BilinForm.toMatrix}(b,\\mathrm{traceForm}(K,L))\\big) \\neq 0$$$

Lean4

theorem trace_surjective [FiniteDimensional K L] [Algebra.IsSeparable K L] : Function.Surjective (Algebra.trace K L) :=
  by
  rw [← LinearMap.range_eq_top]
  apply (IsSimpleOrder.eq_bot_or_eq_top (α := Ideal K) _).resolve_left
  rw [LinearMap.range_eq_bot]
  exact Algebra.trace_ne_zero K L

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