InverseGalois.lean`

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

8 declarations, 6 contain sorry, 0 declared as axioms

Inverse Galois problem

Reference: Wikipedia

Imports (1)

FormalConjecturesUtil

Imported by 1 file

FormalConjectures/Subsets/FC100SolvedSet1.lean

structure InverseGalois.GaloisRealization line 30 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Group
class InverseGalois.IsRealizable Say a group G is realizable over a field K if it is isomorphic to the Galois group of a Galois extension of K line 44 source
Source documentation
Say a group G is realizable over a field K if it is isomorphic to the Galois group of a Galois extension of K

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Group
```
class IsRealizable (K G : Type*) [Field K] [Group G] where
…
```
theorem InverseGalois.inverse_galois_problem The Inverse Galois Problem: every finite group is isomorphic to the Galois group of a Galois extension of the rationals sorry line 53 source
Source documentation
The Inverse Galois Problem: every finite group is isomorphic to the Galois group of a Galois extension of the rationals.

depends on sorry what this means
Statement uses: Fintype, Group, InverseGalois.IsRealizable, Rat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
3 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem inverse_galois_problem {G : Type*} [Fintype G] [Group G] :
    IsRealizable ℚ G := by
…
```
Mentions: InverseGalois.IsRealizable
theorem InverseGalois.inverse_galois_problem.variants.cyclic Every finite cyclic group is realizable sorry line 61 source
Source documentation
Every finite cyclic group is realizable.

depends on sorry what this means
Statement uses: Fintype, Group, IsCyclic, InverseGalois.IsRealizable, Rat
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 inverse_galois_problem.variants.cyclic
    {G : Type*} [Fintype G] [Group G] [IsCyclic G] :
    IsRealizable ℚ G := by
…
```
Mentions: InverseGalois.IsRealizable
theorem InverseGalois.inverse_galois_problem.variants.abelian Every finite abelian group is realizable sorry line 70 source
Source documentation
Every finite abelian group is realizable.

depends on sorry what this means
Statement uses: Fintype, CommGroup, InverseGalois.IsRealizable, Rat
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem inverse_galois_problem.variants.abelian
    {G : Type*} [Fintype G] [CommGroup G] :
    IsRealizable ℚ G := by
…
```
Mentions: InverseGalois.IsRealizable
theorem InverseGalois.inverse_galois_problem.variants.symmetric_group Every finite symmetric group is realizable sorry line 79 source
Source documentation
Every finite symmetric group is realizable.

depends on sorry what this means
Statement uses: Fintype, InverseGalois.IsRealizable, Rat, Equiv
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
5 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem inverse_galois_problem.variants.symmetric_group
    {S : Type*} [Fintype S] :
    IsRealizable ℚ (S ≃ S) := by
…
```
Mentions: InverseGalois.IsRealizable
theorem InverseGalois.inverse_galois_problem.variants.complex_rational_functions Every finite group is realisable over the field of rational functions with complex coefficients sorry line 89 source
Source documentation
Every finite group is realisable over the field of rational functions with complex coefficients.

depends on sorry what this means
Statement uses: Fintype, Group, InverseGalois.IsRealizable, RatFunc, Complex
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
13 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem inverse_galois_problem.variants.complex_rational_functions
    {G : Type*} [Fintype G] [Group G] :
    IsRealizable (RatFunc ℂ) G := by
…
```
Mentions: InverseGalois.IsRealizable
theorem InverseGalois.inverse_galois_problem.variants.complex_function_field Every finite group is realisable over the field of rational functions with coefficients K, where K is any algebraically closed field of characteristic 0 sorry line 99 source
Source documentation
Every finite group is realisable over the field of rational functions with coefficients K, where K is any algebraically closed field of characteristic 0.

depends on sorry what this means
Statement uses: Field, IsAlgClosed, CharZero, Fintype, Group, InverseGalois.IsRealizable, RatFunc
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous, OfNat.ofNat, Nat
17 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem inverse_galois_problem.variants.complex_function_field
    {G K : Type*} [Field K] [IsAlgClosed K] [CharZero K] [Fintype G] [Group G] :
    IsRealizable (RatFunc K) G := by
…
```
Mentions: InverseGalois.IsRealizable