English
For a finite extension K ⊂ L, the following are equivalent: L/K is separable; the trace is nonzero; the trace form is nondegenerate.
Русский
Для конечного расширения K ⊂ L эквивалентны: развиваемость L/K; след не нулевой; форма следа неdegenerate.
LaTeX
$$$\\text{TFAE}([\\text{FiniteDimensional }K,L])\\;:\\; [\\text{Algebra.IsSeparable }K L,\\; \\text{trace}(K L) \\neq 0,\\; (\\text{traceForm}(K,L)).\\text{Nondegenerate}]$$
Lean4
@[stacks 0BIL]
theorem traceForm_nondegenerate_tfae [FiniteDimensional K L] :
[Algebra.IsSeparable K L, Algebra.trace K L ≠ 0, (traceForm K L).Nondegenerate].TFAE :=
by
tfae_have 1 → 3 := fun _ ↦ traceForm_nondegenerate K L
tfae_have 3 → 2 := fun H₁ H₂ ↦ H₁.ne_zero (by ext; simp [H₂])
tfae_have 2 → 1 := not_imp_comm.mp Algebra.trace_eq_zero_of_not_isSeparable
tfae_finish
