Theorems · Theorem · number theory

QuadraticMap.nondegenerate_iff_radical_eq_bot

∀ {R : Type u_1} {M : Type u_2} {P : Type u_4} [inst : AddCommGroup M] [inst_1 : AddCommGroup P] [inst_2 : CommRing R]
  [inst_3 : Module R M] [inst_4 : Module R P] {Q : QuadraticMap R M P} [Invertible 2],
  QuadraticMap.Nondegenerate ↔ Q.radical = ⊥

If 2 is invertible in the coefficient ring, a quadratic map is nondegenerate iff its radical is 0. (Use QuadraticMap.Nondegenerate.radical_eq_bot for the one-way implication in all characteristics.)

Defined in

Mathlib.LinearAlgebra.QuadraticForm.Radical

Cited by

2 results in Mathlib

Foundations

Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AddCommGroupAddCommGroupCommRingModuleModuleInvertible