NoetherProblem.lean`

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Credit: The upstream authors and contributors retain credit for the project and mathematics. lean4-explainer only renders their pinned source and adds the separately disclosed explanation layer.

Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

8 declarations, 5 contain sorry, 0 declared as axioms

Rational_variety

Reference: Wikipedia

Imports (1)

FormalConjecturesUtil

class NoetherProblem.IsRationalExtension A rational field extension is a field extension L/K isomorphic to a field of rational functions (in some arbitrary number of indeterminates.) line 34 source
Source documentation
A rational field extension is a field extension L/K isomorphic to a field of rational functions (in some arbitrary number of indeterminates.)

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Algebra
4 instances, coercions and auxiliary constants left out.
```
class IsRationalExtension (K L ι : Type*)
    [Field K] [Field L] [Algebra K L] where
…
```
theorem NoetherProblem.rationalExtension_empty_index If the index set ι is empty, then IsRationalExtension K L ι means that K, L are isomorphic as K algebras line 42 source
Source documentation
If the index set ι is empty, then IsRationalExtension K L ι means that K, L are isomorphic as K algebras.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Algebra, IsEmpty, NoetherProblem.IsRationalExtension, Nonempty, AlgEquiv
Proof uses: FractionRing, MvPolynomial, Finsupp, Nat, nonZeroDivisors, Classical.choice, NoetherProblem.IsRationalExtension.pure_transcendental, MvPolynomial.isEmptyAlgEquiv, IsFractionRing.fieldEquivOfAlgEquiv, Nonempty.intro, AlgEquiv.trans
61 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem rationalExtension_empty_index (K L ι : Type*) [Field K] [Field L] [Algebra K L] [IsEmpty ι]
    [IsRationalExtension K L ι] :
    Nonempty (L ≃ₐ[K] K) := by
…
```
Mentions: NoetherProblem.IsRationalExtension
def NoetherProblem.HasNoetherProperty Documented declaration: HasNoetherProperty line 58 source
Source documentation
We say that a rational extension L of K in the indeterminates ι has the Noether Property if, for every identification of L with the rational function field K(X_i : i ∈ ι), the fixed field L^H of every group H of K-automorphisms of L permuting the indeterminates X_i is again a rational extension of K. Such a group H is necessarily finite.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Fintype, Algebra, NoetherProblem.IsRationalExtension
Proof uses: AlgEquiv, FractionRing, MvPolynomial, Finsupp, Nat, nonZeroDivisors, Subgroup, Membership.mem, Exists, Equiv.Perm, Eq, MulEquiv, MulEquiv.symm, AlgEquiv.autCongr, IsFractionRing.algEquivOfAlgEquiv, MvPolynomial.renameEquiv, Subtype, IntermediateField, IntermediateField.fixedField
47 instances, coercions and auxiliary constants left out.
```
def HasNoetherProperty (K L ι : Type) [Field K] [Field L] [Fintype ι]
    [Algebra K L] [IsRationalExtension K L ι] : Prop :=
…
```
Mentions: NoetherProblem.IsRationalExtension
theorem NoetherProblem.noether_problem The Noether Problem: let L be the field of rational functions in n indeterminates over K, and let G be a finite group permuting these indeterminates sorry line 73 source
Source documentation
The Noether Problem: let L be the field of rational functions in n indeterminates over K, and let G be a finite group permuting these indeterminates. Is the fixed field L^G a rational extension of K, i.e. does L/K have the Noether property?
Solution: False.

depends on sorry what this means
Statement uses: Iff, False, Field, Fintype, Algebra, NoetherProblem.IsRationalExtension, NoetherProblem.HasNoetherProperty
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
9 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem noether_problem : answer(False) ↔ ∀ (K L ι : Type)
    [Field K] [Field L] [Fintype ι] [Algebra K L] [IsRationalExtension K L ι],
    HasNoetherProperty K L ι := by
…
```
Mentions: NoetherProblem.IsRationalExtension, NoetherProblem.HasNoetherProperty
theorem NoetherProblem.noether_problem.variants.two The Noether problem has a positive solution for groups permuting two indeterminates sorry line 82 source
Source documentation
The Noether problem has a positive solution for groups permuting two indeterminates.

depends on sorry what this means
Statement uses: Field, Fintype, Algebra, NoetherProblem.IsRationalExtension, Eq, Nat, Fintype.card, OfNat.ofNat, NoetherProblem.HasNoetherProperty
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem noether_problem.variants.two {K L ι : Type}
    [Field K] [Field L] [Fintype ι] [Algebra K L]
    [IsRationalExtension K L ι] (hι : Fintype.card ι = 2) :
    HasNoetherProperty K L ι := by
…
```
Mentions: NoetherProblem.IsRationalExtension, NoetherProblem.HasNoetherProperty
theorem NoetherProblem.noether_problem.variants.three The Noether problem has a positive solution for groups permuting three indeterminates sorry line 92 source
Source documentation
The Noether problem has a positive solution for groups permuting three indeterminates.

depends on sorry what this means
Statement uses: Field, Fintype, Algebra, NoetherProblem.IsRationalExtension, Eq, Nat, Fintype.card, OfNat.ofNat, NoetherProblem.HasNoetherProperty
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem noether_problem.variants.three {K L ι : Type}
    [Field K] [Field L] [Fintype ι] [Algebra K L]
    [IsRationalExtension K L ι] (hι : Fintype.card ι = 3) :
    HasNoetherProperty K L ι := by
…
```
Mentions: NoetherProblem.IsRationalExtension, NoetherProblem.HasNoetherProperty
theorem NoetherProblem.noether_problem.variants.four The Noether problem has a positive solution for groups permuting four indeterminates sorry line 102 source
Source documentation
The Noether problem has a positive solution for groups permuting four indeterminates.

depends on sorry what this means
Statement uses: Field, Fintype, Algebra, NoetherProblem.IsRationalExtension, Eq, Nat, Fintype.card, OfNat.ofNat, NoetherProblem.HasNoetherProperty
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem noether_problem.variants.four {K L ι : Type}
    [Field K] [Field L] [Fintype ι] [Algebra K L]
    [IsRationalExtension K L ι] (hι : Fintype.card ι = 4) :
    HasNoetherProperty K L ι := by
…
```
Mentions: NoetherProblem.IsRationalExtension, NoetherProblem.HasNoetherProperty
theorem NoetherProblem.noether_problem.variants.forty_seven One can find a counterexample to the Noether Problem's claim by considering a group permuting the 47 indeterminates of a rational function field sorry line 113 source
Source documentation
One can find a counterexample to the Noether Problem's claim by considering a group permuting the 47 indeterminates of a rational function field.

depends on sorry what this means
Statement uses: Exists, Field, Fintype, Algebra, NoetherProblem.IsRationalExtension, And, Eq, Nat, Fintype.card, OfNat.ofNat, Not, NoetherProblem.HasNoetherProperty
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
10 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem noether_problem.variants.forty_seven :
    ∃ (K L ι : Type)
    (_ :  Field K) (_ : Field L) (_ : Fintype ι) (_ : Algebra K L)
    (_ : IsRationalExtension K L ι),
    Fintype.card ι = 47 ∧ ¬ HasNoetherProperty K L ι := by
…
```
Mentions: NoetherProblem.IsRationalExtension, NoetherProblem.HasNoetherProperty