IsQuaternionAlgebra.lean`

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11 declarations, 0 contain sorry, 0 declared as axioms

Is Quaternion Algebra

Material destined for Mathlib.

Imports (4)

Imported by 2 files

def Minpoly.toAdjoin' line 27 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Ring, Algebra, AlgHom, AdjoinRoot, minpoly, Subtype, Membership.mem, Subalgebra, Algebra.adjoin, Singleton.singleton, Set
Proof uses: AdjoinRoot.liftAlgHom, Algebra.ofId, Subtype.mk
22 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
lemma Minpoly.toAdjoin'_surjective line 33 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Ring, Algebra, Function.Surjective, AdjoinRoot, minpoly, Subtype, Membership.mem, Subalgebra, Algebra.adjoin, Singleton.singleton, Set, AlgHom, Minpoly.toAdjoin'
Proof uses: Eq.mpr, Eq, AlgHom.range, Top.top, id, congrArg, Eq.symm, propext, AlgHom.range_eq_top, LE.le, eq_top_iff, Set.preimage, Algebra.adjoin_adjoin_coe_preimage, Algebra.adjoin_le, Subtype.mk, Exists.intro, RingHom, AlgHom.toRingHom, Polynomial, AdjoinRoot.mk and 10 more
50 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
def AlgEquiv.adjoinSingletonEquivAdjoinRootMinpoly' line 43 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Ring, Algebra, AlgEquiv, Subtype, Membership.mem, Subalgebra, Algebra.adjoin, Singleton.singleton, Set, AdjoinRoot, minpoly
Proof uses: AlgEquiv.symm, AlgEquiv.ofBijective, Minpoly.toAdjoin'
29 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
class IsQuaternionAlgebra IsQuaternionAlgebra F D says that D is a quaternion algebra over the field F, i.e. a four-dimensional central simple F-algebra line 54 source
Source documentation
IsQuaternionAlgebra F D says that D is a quaternion algebra over the field F, i.e. a four-dimensional central simple F-algebra.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Ring, Algebra
3 instances, coercions and auxiliary constants left out.
```
class IsQuaternionAlgebra (F : Type*) [Field F] (D : Type*) [Ring D] [Algebra F D] : Prop where
…
```
instance IsQuaternionAlgebra.(instance at line 65) line 65 source
lemma IsQuaternionAlgebra.finrank_eq_four line 67 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Ring, Algebra, IsQuaternionAlgebra, Eq, Nat, Module.finrank, OfNat.ofNat
Proof uses: Module.finrank_eq_of_rank_eq, IsQuaternionAlgebra.dim_four
14 instances, coercions and auxiliary constants left out.
lemma IsQuaternionAlgebra.not_isMulCommutative line 72 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Ring, Algebra, IsQuaternionAlgebra, Not, IsMulCommutative
Proof uses: dite, False, Eq, Subalgebra, IsMulCommutative.instCommRing, Bot.bot, Top.top, Classical.propDecidable, False.elim, Eq.mp, Eq.trans, Nat, Module.finrank, OfNat.ofNat, congrFun', congrArg, IsQuaternionAlgebra.finrank_eq_four, absurd, Eq.refl, Subalgebra.center and 3 more
63 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma IsQuaternionAlgebra.nomepty_algEquiv_matrix_or_forall_isUnit line 79 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Ring, Algebra, IsQuaternionAlgebra, Or, Nonempty, AlgEquiv, Matrix, Fin, OfNat.ofNat, Nat, Ne, IsUnit
Proof uses: IsArtinianRing, IsArtinianRing.of_finite, Exists, NeZero, DivisionRing, IsSimpleRing.exists_algEquiv_matrix_divisionRing, Eq, Module.finrank, IsQuaternionAlgebra.finrank_eq_four, Nat.casesAuxOn, HMul.hMul, False.elim, Eq.mp, False, Eq.trans, congrFun', congrArg, MulZeroClass.mul_zero, MulZeroClass.zero_mul, HAdd.hAdd and 43 more
118 instances, coercions and auxiliary constants left out.

Tactic trace: 20 steps
Open to load the checked before-and-after goal states.
lemma IsQuaternionAlgebra.Algebra.finrank_adjoin_singleton line 95 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Ring, Algebra, IsIntegral, Eq, Nat, Module.finrank, Subtype, Membership.mem, Subalgebra, Algebra.adjoin, Singleton.singleton, Set, Polynomial.natDegree, minpoly
Proof uses: Eq.mpr, AdjoinRoot, id, congrArg, LinearEquiv.finrank_eq, AlgEquiv.adjoinSingletonEquivAdjoinRootMinpoly', PowerBasis.dim, AdjoinRoot.powerBasis, minpoly.ne_zero, PowerBasis.finrank, AdjoinRoot.powerBasis_dim, Eq.refl
42 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
lemma IsQuaternionAlgebra.natDegree_minpoly_le_two line 103 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Ring, Algebra, IsQuaternionAlgebra, LE.le, Nat, Polynomial.natDegree, minpoly, OfNat.ofNat
Proof uses: Nonempty, AlgEquiv, Matrix, Fin, Ne, IsUnit, Or, IsQuaternionAlgebra.nomepty_algEquiv_matrix_or_forall_isUnit, Eq.mpr, AlgHom, id, Eq, congrArg, Polynomial, Eq.symm, minpoly.algHom_eq, AlgEquiv.injective, LE.le.trans, Matrix.charpoly, Polynomial.natDegree_le_of_dvd and 78 more
160 instances, coercions and auxiliary constants left out.

Tactic trace: 31 steps
Open to load the checked before-and-after goal states.
class IsQuaternionAlgebra.IsTotallyDefinite Documented declaration: IsTotallyDefinite line 143 source
Source documentation
A quaternion algebra D over a number field F is totally definite if D ⊗[F, v] ℝ is isomorphic to the Hamilton quaternions ℍ for all real places v (that is, for all ring homomorphisms F → ℝ).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Field, Ring, Algebra
3 instances, coercions and auxiliary constants left out.
```
class IsTotallyDefinite : Prop where
…
```