UInvariant.lean`

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8 declarations, 0 contain sorry, 0 declared as axioms

Dimensions of anisotropic quadratic forms

The $u$-invariant $u(F)$ of a field $F$ is the largest dimension of an anisotropic quadratic form over $F$, or $\infty$ if there is no largest one [Kaplansky1953, p. 201]; in characteristic $2$ the forms are required to be nondegenerate [EKM2008, §36]. This file defines

QuadraticForm.anisotropicDims F: the set of n such that there is a nondegenerate anisotropic quadratic form on Fin n → F.

The $u$-invariant of F is n exactly when IsGreatest (QuadraticForm.anisotropicDims F) n, and it is infinite exactly when ¬ BddAbove (QuadraticForm.anisotropicDims F). No numeric invariant is defined, so that no convention for the infinite case is needed.

The set contains 0 and 1 for every field. In characteristic not 2 every anisotropic form is nondegenerate (QuadraticMap.Anisotropic.nondegenerate), so the nondegeneracy condition can be dropped (QuadraticForm.mem_anisotropicDims_iff). Over a linearly ordered field the sum of n squares is anisotropic for every n, so the set is all of ℕ (QuadraticForm.anisotropicDims_eq_univ).

References

[Kaplansky1953] I. Kaplansky, Quadratic forms, J. Math. Soc. Japan 5 (1953), 200–207.
[EKM2008] R. Elman, N. Karpenko, A. Merkurjev, The algebraic and geometric theory of quadratic forms, Amer. Math. Soc. Colloq. Publ. 56, 2008.

Imports (1)

Mathlib.LinearAlgebra.QuadraticForm.Radical

Imported by 1 file

FormalConjecturesForMathlib.lean

theorem QuadraticMap.Anisotropic.radical_eq_bot The radical of an anisotropic quadratic form is trivial line 56 source
Source documentation
The radical of an anisotropic quadratic form is trivial.

standard axioms: Quot.sound, propext what this means
Statement uses: Field, AddCommGroup, Module, QuadraticForm, QuadraticMap.Anisotropic, Eq, Submodule, QuadraticMap.radical, Bot.bot
Proof uses: Iff.mpr, Membership.mem, OfNat.ofNat, Submodule.eq_bot_iff, And.left, QuadraticMap, HAdd.hAdd, Iff.mp, And, QuadraticMap.mem_radical_iff'
46 instances, coercions and auxiliary constants left out.
```
theorem Anisotropic.radical_eq_bot (hQ : Q.Anisotropic) : Q.radical = ⊥ :=
…
```
theorem QuadraticMap.Anisotropic.nondegenerate_of_rank_le_one An anisotropic quadratic form on a module of rank at most 1 is nondegenerate line 60 source
Source documentation
An anisotropic quadratic form on a module of rank at most 1 is nondegenerate.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, AddCommGroup, Module, QuadraticForm, QuadraticMap.Anisotropic, LE.le, Cardinal, Module.rank, OfNat.ofNat, QuadraticMap.Nondegenerate
Proof uses: QuadraticMap.Nondegenerate.mk, QuadraticMap.Anisotropic.radical_eq_bot, LE.le.trans, Subtype, Membership.mem, Submodule, LinearMap.ker, LinearMap, RingHom.id, QuadraticMap.polarBilin, Submodule.rank_le
41 instances, coercions and auxiliary constants left out.
```
theorem Anisotropic.nondegenerate_of_rank_le_one (hQ : Q.Anisotropic)
    (hM : Module.rank F M ≤ 1) : Q.Nondegenerate where
…
```
theorem QuadraticMap.Anisotropic.nondegenerate In characteristic not 2, an anisotropic quadratic form is nondegenerate line 66 source
Source documentation
In characteristic not 2, an anisotropic quadratic form is nondegenerate.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, AddCommGroup, Module, QuadraticForm, NeZero, OfNat.ofNat, Nat, QuadraticMap.Anisotropic, QuadraticMap.Nondegenerate
Proof uses: Invertible, invertibleOfNonzero, two_ne_zero, Iff.mpr, Eq, Submodule, QuadraticMap.radical, Bot.bot, QuadraticMap.nondegenerate_iff_radical_eq_bot, QuadraticMap.Anisotropic.radical_eq_bot
53 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
theorem Anisotropic.nondegenerate [NeZero (2 : F)] (hQ : Q.Anisotropic) : Q.Nondegenerate := by
…
```
def QuadraticForm.anisotropicDims The set of n such that there is a nondegenerate anisotropic quadratic form on Fin n → F line 79 source
Source documentation
The set of n such that there is a nondegenerate anisotropic quadratic form on Fin n → F. Its greatest element, when it exists, is the $u$-invariant of F; the set is not bounded above exactly when the $u$-invariant of F is infinite.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Set, Nat
Proof uses: Set.ofPred, Exists, QuadraticForm, Fin, And, QuadraticMap.Anisotropic, QuadraticMap.Nondegenerate
14 instances, coercions and auxiliary constants left out.
```
def anisotropicDims : Set ℕ :=
…
```
theorem QuadraticForm.zero_mem_anisotropicDims line 82 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Membership.mem, Nat, Set, QuadraticForm.anisotropicDims, OfNat.ofNat
Proof uses: QuadraticMap.Anisotropic, Fin, QuadraticForm, Eq, QuadraticMap, Subsingleton.elim, Exists.intro, And, QuadraticMap.Nondegenerate, And.intro, QuadraticMap.Anisotropic.nondegenerate_of_rank_le_one, of_eq_true, LE.le, Cardinal, Module.rank, Eq.trans, True, congrFun', congrArg, rank_subsingleton'
46 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
theorem QuadraticForm.one_mem_anisotropicDims line 86 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Membership.mem, Nat, Set, QuadraticForm.anisotropicDims, OfNat.ofNat
Proof uses: QuadraticMap.Anisotropic, Fin, QuadraticMap.weightedSumSquares, Eq, QuadraticMap, funext, Eq.mpr, id, congrArg, Fin.fin_one_eq_zero, Eq.mp, Eq.trans, HMul.hMul, congrFun', Finset.sum, Finset.univ, HSMul.hSMul, QuadraticMap.weightedSumSquares_apply, Finset.sum_congr, Eq.refl and 28 more
78 instances, coercions and auxiliary constants left out.

Tactic trace: 5 steps
Open to load the checked before-and-after goal states.
theorem QuadraticForm.mem_anisotropicDims_iff In characteristic not 2, the nondegeneracy condition in anisotropicDims is automatic line 96 source
Source documentation
In characteristic not 2, the nondegeneracy condition in anisotropicDims is automatic.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, NeZero, OfNat.ofNat, Nat, Iff, Membership.mem, Set, QuadraticForm.anisotropicDims, Exists, QuadraticForm, Fin, QuadraticMap.Anisotropic
Proof uses: Iff.intro, QuadraticMap.Nondegenerate, Exists.intro, And, And.intro, QuadraticMap.Anisotropic.nondegenerate
47 instances, coercions and auxiliary constants left out.
```
theorem mem_anisotropicDims_iff [NeZero (2 : F)] {n : ℕ} :
    n ∈ anisotropicDims F ↔ ∃ Q : QuadraticForm F (Fin n → F), Q.Anisotropic :=
…
```
Mentions: QuadraticForm.anisotropicDims
theorem QuadraticForm.anisotropicDims_eq_univ Over a linearly ordered field, the sum of n squares is anisotropic for every n, so every n is the dimension of an anisotropic quadratic form line 102 source
Source documentation
Over a linearly ordered field, the sum of n squares is anisotropic for every n, so every n is the dimension of an anisotropic quadratic form.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, LinearOrder, IsStrictOrderedRing, Eq, Set, Nat, QuadraticForm.anisotropicDims, Set.univ
Proof uses: Set.eq_univ_of_forall, Iff.mpr, Membership.mem, Exists, QuadraticForm, Fin, QuadraticMap.Anisotropic, QuadraticForm.mem_anisotropicDims_iff, Exists.intro, QuadraticMap.weightedSumSquares, OfNat.ofNat, QuadraticMap, funext, Iff.mp, HMul.hMul, mul_self_eq_zero, Eq.mp, Finset.sum, Finset.univ, Finset and 12 more
69 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem anisotropicDims_eq_univ [LinearOrder F] [IsStrictOrderedRing F] :
    anisotropicDims F = Set.univ :=
…
```
Mentions: QuadraticForm.anisotropicDims