Field

Field type class. While algebra already provides one, we provide one in Spire that integrates with the commutative ring hierarchy, in particular GCDRing and EuclideanRing.

On a field, all nonzero elements are invertible, so the remainder of the division is always 0. The Euclidean function can take an arbitrary value on nonzero elements (it is undefined for zero); for compatibility with the degree of polynomials, we use the constant 0.

The GCD and LCM are defined up to a unit; on a field, it means that either the GCD or LCM can be fixed arbitrarily. Some conventions with consistent defaults are provided in the spire.algebra.Field companion object.

Linear Supertypes

EuclideanRing[A], GCDRing[A], algebra.ring.Field[A], MultiplicativeCommutativeGroup[A], algebra.ring.MultiplicativeGroup[A], CommutativeRing[A], CommutativeRng[A], CommutativeRig[A], MultiplicativeCommutativeMonoid[A], CommutativeSemiring[A], MultiplicativeCommutativeSemigroup[A], algebra.ring.Ring[A], algebra.ring.Rng[A], AdditiveCommutativeGroup[A], algebra.ring.AdditiveGroup[A], algebra.ring.Rig[A], algebra.ring.MultiplicativeMonoid[A], algebra.ring.Semiring[A], algebra.ring.MultiplicativeSemigroup[A], AdditiveCommutativeMonoid[A], AdditiveCommutativeSemigroup[A], algebra.ring.AdditiveMonoid[A], algebra.ring.AdditiveSemigroup[A], Serializable, Serializable, Any

Known Subclasses

BigDecimalAlgebra, BigDecimalIsField, DistField, DoubleAlgebra, DoubleIsField, FieldOfFractionsGCD, FloatAlgebra, FloatIsField, Fractional, RealAlgebra, RealIsFractional, WithDefaultGCD

Abstract Value Members

abstract def div(x: A, y: A): A

Definition Classes
MultiplicativeGroup
abstract def gcd(a: A, b: A)(implicit ev: Eq[A]): A

Definition Classes
GCDRing
abstract def getClass(): Class[_]

Definition Classes
Any
abstract def lcm(a: A, b: A)(implicit ev: Eq[A]): A

Definition Classes
GCDRing
abstract def negate(x: A): A

Definition Classes
AdditiveGroup
abstract def one: A

Definition Classes
MultiplicativeMonoid
abstract def plus(x: A, y: A): A

Definition Classes
AdditiveSemigroup
abstract def times(x: A, y: A): A

Definition Classes
MultiplicativeSemigroup
abstract def zero: A

Definition Classes
AdditiveMonoid

Concrete Value Members

final def !=(arg0: Any): Boolean

Definition Classes
Any
final def ##(): Int

Definition Classes
Any
final def ==(arg0: Any): Boolean

Definition Classes
Any
def additive: CommutativeGroup[A]

Definition Classes
AdditiveCommutativeGroup → AdditiveCommutativeMonoid → AdditiveCommutativeSemigroup → AdditiveGroup → AdditiveMonoid → AdditiveSemigroup
final def asInstanceOf[T0]: T0

Definition Classes
Any
def emod(a: A, b: A): A

Definition Classes
Field → EuclideanRing
def equals(arg0: Any): Boolean

Definition Classes
Any
def equot(a: A, b: A): A

Definition Classes
Field → EuclideanRing
def equotmod(a: A, b: A): (A, A)

Definition Classes
Field → EuclideanRing
def euclideanFunction(a: A): BigInt

Definition Classes
Field → EuclideanRing
def fromBigInt(n: BigInt): A

Definition Classes
Ring
def fromDouble(a: Double): A

Definition Classes
Field
def fromInt(n: Int): A

Definition Classes
Ring
def hashCode(): Int

Definition Classes
Any
final def isInstanceOf[T0]: Boolean

Definition Classes
Any
def isOne(a: A)(implicit ev: algebra.Eq[A]): Boolean

Definition Classes
MultiplicativeMonoid
def isZero(a: A)(implicit ev: algebra.Eq[A]): Boolean

Definition Classes
AdditiveMonoid
def minus(x: A, y: A): A

Definition Classes
AdditiveGroup
def multiplicative: CommutativeGroup[A]

Definition Classes
MultiplicativeCommutativeGroup → MultiplicativeCommutativeMonoid → MultiplicativeCommutativeSemigroup → MultiplicativeGroup → MultiplicativeMonoid → MultiplicativeSemigroup
def positivePow(a: A, n: Int): A

Attributes
protected[this]
Definition Classes
MultiplicativeSemigroup
def positiveSumN(a: A, n: Int): A

Attributes
protected[this]
Definition Classes
AdditiveSemigroup
def pow(a: A, n: Int): A

Definition Classes
MultiplicativeGroup → MultiplicativeMonoid → MultiplicativeSemigroup
def product(as: TraversableOnce[A]): A

Definition Classes
MultiplicativeMonoid
def reciprocal(x: A): A

Definition Classes
MultiplicativeGroup
def sum(as: TraversableOnce[A]): A

Definition Classes
AdditiveMonoid
def sumN(a: A, n: Int): A

Definition Classes
AdditiveGroup → AdditiveMonoid → AdditiveSemigroup
def toString(): String

Definition Classes
Any
def tryProduct(as: TraversableOnce[A]): Option[A]

Definition Classes
MultiplicativeMonoid → MultiplicativeSemigroup
def trySum(as: TraversableOnce[A]): Option[A]

Definition Classes
AdditiveMonoid → AdditiveSemigroup

Inherited from EuclideanRing[A]

Inherited from GCDRing[A]

Inherited from algebra.ring.Field[A]

Inherited from MultiplicativeCommutativeGroup[A]

Inherited from algebra.ring.MultiplicativeGroup[A]

Inherited from CommutativeRing[A]

Inherited from CommutativeRng[A]

Related Docs: object Field | package algebra

trait Field[A] extends AlgebraField[A] with EuclideanRing[A]

Abstract Value Members

abstract def div(x: A, y: A): A

abstract def gcd(a: A, b: A)(implicit ev: Eq[A]): A

abstract def getClass(): Class[_]

abstract def lcm(a: A, b: A)(implicit ev: Eq[A]): A

abstract def negate(x: A): A

abstract def one: A

abstract def plus(x: A, y: A): A

abstract def times(x: A, y: A): A

abstract def zero: A

Concrete Value Members

final def !=(arg0: Any): Boolean

final def ##(): Int

final def ==(arg0: Any): Boolean

def additive: CommutativeGroup[A]

final def asInstanceOf[T0]: T0

def emod(a: A, b: A): A

def equals(arg0: Any): Boolean

def equot(a: A, b: A): A

def equotmod(a: A, b: A): (A, A)

def euclideanFunction(a: A): BigInt

def fromBigInt(n: BigInt): A

def fromDouble(a: Double): A

def fromInt(n: Int): A

def hashCode(): Int

final def isInstanceOf[T0]: Boolean

def isOne(a: A)(implicit ev: algebra.Eq[A]): Boolean

def isZero(a: A)(implicit ev: algebra.Eq[A]): Boolean

def minus(x: A, y: A): A

def multiplicative: CommutativeGroup[A]

def positivePow(a: A, n: Int): A

def positiveSumN(a: A, n: Int): A

def pow(a: A, n: Int): A

def product(as: TraversableOnce[A]): A

def reciprocal(x: A): A

def sum(as: TraversableOnce[A]): A

def sumN(a: A, n: Int): A

def toString(): String

def tryProduct(as: TraversableOnce[A]): Option[A]

def trySum(as: TraversableOnce[A]): Option[A]

Inherited from EuclideanRing[A]

Inherited from GCDRing[A]

Inherited from algebra.ring.Field[A]

Inherited from MultiplicativeCommutativeGroup[A]

Inherited from algebra.ring.MultiplicativeGroup[A]

Inherited from CommutativeRing[A]

Inherited from CommutativeRng[A]

Inherited from CommutativeRig[A]

Inherited from MultiplicativeCommutativeMonoid[A]

Inherited from CommutativeSemiring[A]

Inherited from MultiplicativeCommutativeSemigroup[A]

Inherited from algebra.ring.Ring[A]

Inherited from algebra.ring.Rng[A]

Inherited from AdditiveCommutativeGroup[A]

Inherited from algebra.ring.AdditiveGroup[A]

Inherited from algebra.ring.Rig[A]

Inherited from algebra.ring.MultiplicativeMonoid[A]

Inherited from algebra.ring.Semiring[A]

Inherited from algebra.ring.MultiplicativeSemigroup[A]

Inherited from AdditiveCommutativeMonoid[A]

Inherited from AdditiveCommutativeSemigroup[A]

Inherited from algebra.ring.AdditiveMonoid[A]

Inherited from algebra.ring.AdditiveSemigroup[A]

Inherited from Serializable

Inherited from Serializable

Inherited from Any

Ungrouped