English
The trace map from a finite field F to its prime field ZMod(ringChar F) is nondegenerate: for any nonzero a ∈ F there exists b ∈ F with trace(a b) ≠ 0.
Русский
Следовая карта from конечного поля F к его простому полю ZMod(ringChar F) не вырождается: для любого не нуля в F найдётся элемент b ∈ F такой, что trace(a b) ≠ 0.
LaTeX
$$$$ \text{There exists } b \in F \text{ with } \\mathrm{trace}(a b) \\neq 0 \text{ for } a \\neq 0. $$$$
Lean4
/-- The trace map from a finite field to its prime field is nongedenerate. -/
theorem trace_to_zmod_nondegenerate (F : Type*) [Field F] [Finite F] [Algebra (ZMod (ringChar F)) F] {a : F}
(ha : a ≠ 0) : ∃ b : F, Algebra.trace (ZMod (ringChar F)) F (a * b) ≠ 0 :=
by
haveI : Fact (ringChar F).Prime := ⟨CharP.char_is_prime F _⟩
have htr := traceForm_nondegenerate (ZMod (ringChar F)) F a
simp_rw [Algebra.traceForm_apply] at htr
by_contra! hf
exact ha (htr hf)
