# Documentation

Mathlib.Algebra.Field.Defs

# Division (semi)rings and (semi)fields #

This file introduces fields and division rings (also known as skewfields) and proves some basic statements about them. For a more extensive theory of fields, see the FieldTheory folder.

## Main definitions #

• DivisionSemiring: Nontrivial semiring with multiplicative inverses for nonzero elements.
• DivisionRing: : Nontrivial ring with multiplicative inverses for nonzero elements.
• Semifield: Commutative division semiring.
• Field: Commutative division ring.
• IsField: Predicate on a (semi)ring that it is a (semi)field, i.e. that the multiplication is commutative, that it has more than one element and that all non-zero elements have a multiplicative inverse. In contrast to Field, which contains the data of a function associating to an element of the field its multiplicative inverse, this predicate only assumes the existence and can therefore more easily be used to e.g. transfer along ring isomorphisms.

## Implementation details #

By convention 0⁻¹ = 0 in a field or division ring. This is due to the fact that working with total functions has the advantage of not constantly having to check that x ≠ 0 when writing x⁻¹. With this convention in place, some statements like (a + b) * c⁻¹ = a * c⁻¹ + b * c⁻¹ still remain true, while others like the defining property a * a⁻¹ = 1 need the assumption a ≠ 0. If you are a beginner in using Lean and are confused by that, you can read more about why this convention is taken in Kevin Buzzard's blogpost

A division ring or field is an example of a GroupWithZero. If you cannot find a division ring / field lemma that does not involve +, you can try looking for a GroupWithZero lemma instead.

## Tags #

field, division ring, skew field, skew-field, skewfield

def Rat.castRec {K : Type u_3} [] [] [Mul K] [Inv K] :
K

The default definition of the coercion (↑(a : ℚ) : K) for a division ring K is defined as (a / b : K) = (a : K) * (b : K)⁻¹. Use coe instead of Rat.castRec for better definitional behaviour.

Instances For
def qsmulRec {K : Type u_3} (coe : K) [Mul K] (a : ) (x : K) :
K

The default definition of the scalar multiplication (a : ℚ) • (x : K) for a division ring K is given by a • x = (↑ a) * x. Use (a : ℚ) • (x : K) instead of qsmulRec for better definitional behaviour.

Instances For
class DivisionSemiring (α : Type u_4) extends , , , :
Type u_4
• add_assoc : ∀ (a b c : α), a + b + c = a + (b + c)
• zero : α
• zero_add : ∀ (a : α), 0 + a = a
• add_zero : ∀ (a : α), a + 0 = a
• nsmul : αα
• nsmul_zero : ∀ (x : α), = 0
• nsmul_succ : ∀ (n : ) (x : α), AddMonoid.nsmul (n + 1) x = x +
• add_comm : ∀ (a b : α), a + b = b + a
• mul : ααα
• left_distrib : ∀ (a b c : α), a * (b + c) = a * b + a * c
• right_distrib : ∀ (a b c : α), (a + b) * c = a * c + b * c
• zero_mul : ∀ (a : α), 0 * a = 0
• mul_zero : ∀ (a : α), a * 0 = 0
• mul_assoc : ∀ (a b c : α), a * b * c = a * (b * c)
• one : α
• one_mul : ∀ (a : α), 1 * a = a
• mul_one : ∀ (a : α), a * 1 = a
• natCast : α
• natCast_zero :
• natCast_succ : ∀ (n : ), NatCast.natCast (n + 1) =
• npow : αα
• npow_zero : ∀ (x : α), = 1
• npow_succ : ∀ (n : ) (x : α), Semiring.npow (n + 1) x = x *
• inv : αα
• div : ααα
• div_eq_mul_inv : ∀ (a b : α), a / b = a * b⁻¹

a / b := a * b⁻¹

• zpow : αα

The power operation: a ^ n = a * ··· * a; a ^ (-n) = a⁻¹ * ··· a⁻¹ (n times)

• zpow_zero' : ∀ (a : α),

a ^ 0 = 1

• zpow_succ' : ∀ (n : ) (a : α), = a *

a ^ (n + 1) = a * a ^ n

• zpow_neg' : ∀ (n : ) (a : α), = (DivisionSemiring.zpow (↑()) a)⁻¹

a ^ -(n + 1) = (a ^ (n + 1))⁻¹

• exists_pair_ne : x y, x y
• inv_zero : 0⁻¹ = 0

The inverse of 0 in a group with zero is 0.

• mul_inv_cancel : ∀ (a : α), a 0a * a⁻¹ = 1

Every nonzero element of a group with zero is invertible.

A DivisionSemiring is a Semiring with multiplicative inverses for nonzero elements.

Instances
class DivisionRing (K : Type u) extends , , , , :
• add_assoc : ∀ (a b c : K), a + b + c = a + (b + c)
• zero : K
• zero_add : ∀ (a : K), 0 + a = a
• add_zero : ∀ (a : K), a + 0 = a
• nsmul : KK
• nsmul_zero : ∀ (x : K), = 0
• nsmul_succ : ∀ (n : ) (x : K), AddMonoid.nsmul (n + 1) x = x +
• add_comm : ∀ (a b : K), a + b = b + a
• mul : KKK
• left_distrib : ∀ (a b c : K), a * (b + c) = a * b + a * c
• right_distrib : ∀ (a b c : K), (a + b) * c = a * c + b * c
• zero_mul : ∀ (a : K), 0 * a = 0
• mul_zero : ∀ (a : K), a * 0 = 0
• mul_assoc : ∀ (a b c : K), a * b * c = a * (b * c)
• one : K
• one_mul : ∀ (a : K), 1 * a = a
• mul_one : ∀ (a : K), a * 1 = a
• natCast : K
• natCast_zero :
• natCast_succ : ∀ (n : ), NatCast.natCast (n + 1) =
• npow : KK
• npow_zero : ∀ (x : K), = 1
• npow_succ : ∀ (n : ) (x : K), Semiring.npow (n + 1) x = x *
• neg : KK
• sub : KKK
• sub_eq_add_neg : ∀ (a b : K), a - b = a + -b
• zsmul : KK
• zsmul_zero' : ∀ (a : K), = 0
• zsmul_succ' : ∀ (n : ) (a : K), Ring.zsmul () a = a + Ring.zsmul () a
• zsmul_neg' : ∀ (n : ) (a : K), = -Ring.zsmul (↑()) a
• add_left_neg : ∀ (a : K), -a + a = 0
• intCast : K
• intCast_ofNat : ∀ (n : ), = n
• intCast_negSucc : ∀ (n : ), = -↑(n + 1)
• inv : KK
• div : KKK
• div_eq_mul_inv : ∀ (a b : K), a / b = a * b⁻¹

a / b := a * b⁻¹

• zpow : KK

The power operation: a ^ n = a * ··· * a; a ^ (-n) = a⁻¹ * ··· a⁻¹ (n times)

• zpow_zero' : ∀ (a : K), = 1

a ^ 0 = 1

• zpow_succ' : ∀ (n : ) (a : K), = a *

a ^ (n + 1) = a * a ^ n

• zpow_neg' : ∀ (n : ) (a : K), = (DivisionRing.zpow (↑()) a)⁻¹

a ^ -(n + 1) = (a ^ (n + 1))⁻¹

• exists_pair_ne : x y, x y
• ratCast : K
• mul_inv_cancel : ∀ (a : K), a 0a * a⁻¹ = 1

For a nonzero a, a⁻¹ is a right multiplicative inverse.

• inv_zero : 0⁻¹ = 0

We define the inverse of 0 to be 0.

• ratCast_mk : ∀ (a : ) (b : ) (h1 : b 0) (h2 : ), ↑(Rat.mk' a b) = a * (b)⁻¹

However ratCast is defined, propositionally it must be equal to a * b⁻¹.

• qsmul : KK

Multiplication by a rational number.

• qsmul_eq_mul' : ∀ (a : ) (x : K), = a * x

However qsmul is defined, propositionally it must be equal to multiplication by ratCast.

A DivisionRing is a Ring with multiplicative inverses for nonzero elements.

An instance of DivisionRing K includes maps ratCast : ℚ → K and qsmul : ℚ → K → K. If the division ring has positive characteristic p, we define ratCast (1 / p) = 1 / 0 = 0 for consistency with our division by zero convention. The fields ratCast and qsmul are needed to implement the algebraRat [DivisionRing K] : Algebra ℚ K instance, since we need to control the specific definitions for some special cases of K (in particular K = ℚ itself). See also Note [forgetful inheritance].

Instances
class Semifield (α : Type u_4) extends , , , :
Type u_4
• add_assoc : ∀ (a b c : α), a + b + c = a + (b + c)
• zero : α
• zero_add : ∀ (a : α), 0 + a = a
• add_zero : ∀ (a : α), a + 0 = a
• nsmul : αα
• nsmul_zero : ∀ (x : α), = 0
• nsmul_succ : ∀ (n : ) (x : α), AddMonoid.nsmul (n + 1) x = x +
• add_comm : ∀ (a b : α), a + b = b + a
• mul : ααα
• left_distrib : ∀ (a b c : α), a * (b + c) = a * b + a * c
• right_distrib : ∀ (a b c : α), (a + b) * c = a * c + b * c
• zero_mul : ∀ (a : α), 0 * a = 0
• mul_zero : ∀ (a : α), a * 0 = 0
• mul_assoc : ∀ (a b c : α), a * b * c = a * (b * c)
• one : α
• one_mul : ∀ (a : α), 1 * a = a
• mul_one : ∀ (a : α), a * 1 = a
• natCast : α
• natCast_zero :
• natCast_succ : ∀ (n : ), NatCast.natCast (n + 1) =
• npow : αα
• npow_zero : ∀ (x : α), = 1
• npow_succ : ∀ (n : ) (x : α), Semiring.npow (n + 1) x = x *
• mul_comm : ∀ (a b : α), a * b = b * a
• inv : αα
• div : ααα
• div_eq_mul_inv : ∀ (a b : α), a / b = a * b⁻¹

a / b := a * b⁻¹

• zpow : αα

The power operation: a ^ n = a * ··· * a; a ^ (-n) = a⁻¹ * ··· a⁻¹ (n times)

• zpow_zero' : ∀ (a : α), = 1

a ^ 0 = 1

• zpow_succ' : ∀ (n : ) (a : α), Semifield.zpow () a = a *

a ^ (n + 1) = a * a ^ n

• zpow_neg' : ∀ (n : ) (a : α), = (Semifield.zpow (↑()) a)⁻¹

a ^ -(n + 1) = (a ^ (n + 1))⁻¹

• exists_pair_ne : x y, x y
• inv_zero : 0⁻¹ = 0

The inverse of 0 in a group with zero is 0.

• mul_inv_cancel : ∀ (a : α), a 0a * a⁻¹ = 1

Every nonzero element of a group with zero is invertible.

A Semifield is a CommSemiring with multiplicative inverses for nonzero elements.

Instances
class Field (K : Type u) extends , , , , :
• add_assoc : ∀ (a b c : K), a + b + c = a + (b + c)
• zero : K
• zero_add : ∀ (a : K), 0 + a = a
• add_zero : ∀ (a : K), a + 0 = a
• nsmul : KK
• nsmul_zero : ∀ (x : K), = 0
• nsmul_succ : ∀ (n : ) (x : K), AddMonoid.nsmul (n + 1) x = x +
• add_comm : ∀ (a b : K), a + b = b + a
• mul : KKK
• left_distrib : ∀ (a b c : K), a * (b + c) = a * b + a * c
• right_distrib : ∀ (a b c : K), (a + b) * c = a * c + b * c
• zero_mul : ∀ (a : K), 0 * a = 0
• mul_zero : ∀ (a : K), a * 0 = 0
• mul_assoc : ∀ (a b c : K), a * b * c = a * (b * c)
• one : K
• one_mul : ∀ (a : K), 1 * a = a
• mul_one : ∀ (a : K), a * 1 = a
• natCast : K
• natCast_zero :
• natCast_succ : ∀ (n : ), NatCast.natCast (n + 1) =
• npow : KK
• npow_zero : ∀ (x : K), = 1
• npow_succ : ∀ (n : ) (x : K), Semiring.npow (n + 1) x = x *
• neg : KK
• sub : KKK
• sub_eq_add_neg : ∀ (a b : K), a - b = a + -b
• zsmul : KK
• zsmul_zero' : ∀ (a : K), = 0
• zsmul_succ' : ∀ (n : ) (a : K), Ring.zsmul () a = a + Ring.zsmul () a
• zsmul_neg' : ∀ (n : ) (a : K), = -Ring.zsmul (↑()) a
• add_left_neg : ∀ (a : K), -a + a = 0
• intCast : K
• intCast_ofNat : ∀ (n : ), = n
• intCast_negSucc : ∀ (n : ), = -↑(n + 1)
• mul_comm : ∀ (a b : K), a * b = b * a
• inv : KK
• div : KKK
• div_eq_mul_inv : ∀ (a b : K), a / b = a * b⁻¹

a / b := a * b⁻¹

• zpow : KK

The power operation: a ^ n = a * ··· * a; a ^ (-n) = a⁻¹ * ··· a⁻¹ (n times)

• zpow_zero' : ∀ (a : K), = 1

a ^ 0 = 1

• zpow_succ' : ∀ (n : ) (a : K), Field.zpow () a = a * Field.zpow () a

a ^ (n + 1) = a * a ^ n

• zpow_neg' : ∀ (n : ) (a : K), = (Field.zpow (↑()) a)⁻¹

a ^ -(n + 1) = (a ^ (n + 1))⁻¹

• exists_pair_ne : x y, x y
• ratCast : K
• mul_inv_cancel : ∀ (a : K), a 0a * a⁻¹ = 1

For a nonzero a, a⁻¹ is a right multiplicative inverse.

• inv_zero : 0⁻¹ = 0

We define the inverse of 0 to be 0.

• ratCast_mk : ∀ (a : ) (b : ) (h1 : b 0) (h2 : ), ↑(Rat.mk' a b) = a * (b)⁻¹

However ratCast is defined, propositionally it must be equal to a * b⁻¹.

• qsmul : KK

Multiplication by a rational number.

• qsmul_eq_mul' : ∀ (a : ) (x : K), = a * x

However qsmul is defined, propositionally it must be equal to multiplication by ratCast.

A Field is a CommRing with multiplicative inverses for nonzero elements.

An instance of Field K includes maps ratCast : ℚ → K and qsmul : ℚ → K → K. If the field has positive characteristic p, we define ratCast (1 / p) = 1 / 0 = 0 for consistency with our division by zero convention. The fields ratCast and qsmul are needed to implement the algebraRat [DivisionRing K] : Algebra ℚ K instance, since we need to control the specific definitions for some special cases of K (in particular K = ℚ itself). See also Note [forgetful inheritance].

Instances
theorem Rat.cast_mk' {K : Type u_3} [] (a : ) (b : ) (h1 : b 0) (h2 : ) :
↑(Rat.mk' a b) = a * (b)⁻¹
theorem Rat.cast_def {K : Type u_3} [] (r : ) :
r = r.num / r.den
instance Rat.smulDivisionRing {K : Type u_3} [] :
theorem Rat.smul_def {K : Type u_3} [] (a : ) (x : K) :
a x = a * x
@[simp]
theorem Rat.smul_one_eq_coe (A : Type u_4) [] (m : ) :
m 1 = m
instance DivisionRing.toOfScientific {K : Type u_3} [] :
instance Field.toSemifield {K : Type u_3} [] :