Factory.Data.QuotientRing

AUTHOR

Dr. Alistair Ward

DESCRIPTION

• Describes a Quotient Ring; https://en.wikipedia.org/wiki/Quotient_ring.
• This is a ring composed from a residue-class resulting from modular division.

Type-classes

class Ring q => QuotientRing q where

Defines a sub-class of Ring, in which division is implemented.

Minimal complete definition

quotRem'

Methods

quotRem' :: q -> q -> (q, q)

Instances

(Eq c, Fractional c, Num e, Ord e) => QuotientRing (Polynomial c e)

Defines the ability to divide polynomials.

Methods

quotRem' :: Polynomial c e -> Polynomial c e -> (Polynomial c e, Polynomial c e)

(Eq c, Num c, Num e, Ord e, Show c, Show e) => QuotientRing (MonicPolynomial c e)

Methods

quotRem' :: MonicPolynomial c e -> MonicPolynomial c e -> (MonicPolynomial c e, MonicPolynomial c e)

Functions

quot'

Arguments

:: QuotientRing q

=> q

Numerator.

-> q

Denominator.

-> q

Returns the quotient, after division of the two specified QuotientRings.

rem'

Arguments

:: QuotientRing q

=> q

Numerator.

-> q

Denominator.

-> q

Returns the remainder, after division of the two specified QuotientRings.

Predicates

areCongruentModulo

Arguments

:: (Eq q, QuotientRing q)

=> q

LHS.

-> q

RHS.

-> q

Modulus.

-> Bool

• True if the two specified QuotientRings are congruent in modulo-arithmetic, where the modulus is a third QuotientRing.
• http://www.usna.edu/Users/math/wdj/book/node74.html.

isDivisibleBy

Arguments

:: (Eq q, QuotientRing q)

=> q

Numerator.

-> q

Denominator.

-> Bool

True if the second operand divides the first.