field

Field

Field

Mathematically a Field is a Commutative Ring all whose non-zero elements have multiplicative inverses.

By convention one != zero, otherwise the algebra consists of just one unique element.

Important

Contract: Field initializer parameters must have

• add closed, commutative and associative on reps

• mult closed, commutative and associative on reps

• one an identity on reps, rep*one == rep == one*rep

• zero an identity on reps, rep+zero == rep == zero+rep

• inv is the mult inverse function on all non-zero reps

• negate function to negate all proper rep values

• invert function to invert all proper rep values

• zero != one (by convention)

class boring_math.abstract_algebra.algebras.field.Field

Bases: CommutativeRing, Generic

Parameters:

• mult – Closed associative function reps.

• add – Closed commutative and associative function reps.

• one – Representation for multiplicative identity.

• zero – Representation for additive identity.

• negate – Function mapping element representation to the representation of corresponding negated element.

• invert – Function mapping non-zero element representations to their multiplicative inverses.

• narrow – Narrow the rep type, many to one function, like choosing an element from a coset of a group,

__init__(mult: Callable[[H, H], H], add: Callable[[H, H], H], one: H, zero: H, negate: Callable[[H], H], invert: Callable[[H], H], narrow: Callable[[H], H] | None = None)

Parameters:

• mult – Closed associative function reps.

• add – Closed commutative and associative function reps.

• one – Representation for multiplicative identity.

• zero – Representation for additive identity.

• negate – Function mapping element representation to the representation of corresponding negated element.

• invert – Function mapping non-zero element representations to their multiplicative inverses.

• narrow – Narrow the rep type, many to one function, like choosing an element from a coset of a group,

class boring_math.abstract_algebra.algebras.field.FieldElement

Bases: CommutativeRingElement, Generic

__init__(rep: H, algebra: Field[H]) → None

__str__() → str

Returns:

str(self) = FieldElement<rep>

__pow__(n: int) → Self

Raise the element to the int power of n.

Parameters:

n – The int power to raise the element to.

Returns:

The element (or its inverse) raised to an int power.

Raises:

• ValueError – If algebra is not multiplicative.

• ValueError – If algebra does not have a multiplicative identity element.

• ValueError – If algebra does not have multiplicative inverses.

__truediv__(right: Self) → Self

Divide self by right.

__add__(right: Self) → Self

Add two elements of the same concrete algebra together.

Parameters:

other – Another element within the same algebra.

Returns:

The sum self + other.

Raises:

• ValueError – If self and other are same type but different concrete algebras.

• TypeError – If Addition not defined on the algebra of the elements.

• TypeError – If self and right are different types.

__mul__(right: object) → Self

Left multiplication for * operator.

Parameters:

right – Object left should be an element of same concrete algebra or an int.

Returns:

Algebra product, the sum of the element right times, or NotImplemented.

Raises:

• ValueError – If either right not an element of the same concrete algebra or right: int < 0.

• TypeError – Multiplication nor defined on the algebra that self is a member of.

__neg__() → Self

Negate the element.

Returns:

The unique additive inverse element to self.

Raises:

ValueError – If algebra fails to have additive inverses.

__radd__(left: Self) → Self

When left side of addition does not know how to add right side.

Parameters:

other – Left side of the addition.

Returns:

Never returns, otherwise left.__add__(right) would have worked.

Raises:

TypeError – When right side does not know how to add the left side to itself.

__rmul__(left: object) → Self

Right multiplication for * operator.

Parameters:

left – Object left should be an int.

Returns:

The sum of the element left times.

Raises:

TypeError – If object on left does not act on object on right