Algebra.PrincipalIdealDomain

A principal ideal domain is a ring in which every ideal (the set of multiples of some generating set of elements) is principal: That is, every element can be written as the multiple of some generating element. gcd a b gives a generator for the ideal generated by a and b. The algorithm above works whenever mod x y is smaller (in a suitable sense) than both x and y; otherwise the algorithm may run forever.

Laws:

   divides x (lcm x y)
   x `gcd` (y `gcd` z) == (x `gcd` y) `gcd` z
   gcd x y * z == gcd (x*z) (y*z)
   gcd x y * lcm x y == x * y

(etc: canonical)

Minimal definition: * nothing, if the standard Euclidean algorithm work * if extendedGCD is implemented customly, gcd and lcm make use of it

Methods

extendedGCD :: a -> a -> (a, (a, a))

Compute the greatest common divisor and solve a respective Diophantine equation.

   (g,(a,b)) = extendedGCD x y ==>
        g==a*x+b*y   &&  g == gcd x y

TODO: This method is not appropriate for the PID class, because there are rings like the one of the multivariate polynomials, where for all x and y greatest common divisors of x and y exist, but they cannot be represented as a linear combination of x and y. TODO: The definition of extendedGCD does not return the canonical associate.

gcd :: a -> a -> a

The Greatest Common Divisor is defined by:

   gcd x y == gcd y x
   divides z x && divides z y ==> divides z (gcd x y)   (specification)
   divides (gcd x y) x

lcm :: a -> a -> a

Least common multiple

Instances

C Int

C Integer

(C a, C a) => C (T a)

(Ord a, C a, C a) => C (T a)

extendedGCD :: C a => a -> a -> (a, (a, a))

Compute the greatest common divisor and solve a respective Diophantine equation.

   (g,(a,b)) = extendedGCD x y ==>
        g==a*x+b*y   &&  g == gcd x y

gcd :: C a => a -> a -> a

The Greatest Common Divisor is defined by:

   gcd x y == gcd y x
   divides z x && divides z y ==> divides z (gcd x y)   (specification)
   divides (gcd x y) x

lcm :: C a => a -> a -> a

Least common multiple

Standard implementations for instances

euclid :: (C a, C a) => (a -> a -> a) -> a -> a -> a

extendedEuclid :: (C a, C a) => (a -> a -> (a, a)) -> a -> a -> (a, (a, a))

Algorithms

extendedGCDMulti :: C a => [a] -> (a, [a])

Compute the greatest common divisor for multiple numbers by repeated application of the two-operand-gcd.

diophantine :: C a => a -> a -> a -> Maybe (a, a)

A variant with small coefficients.

Just (a,b) = diophantine z x y means a*x+b*y = z. It is required that gcd(y,z) divides x.

diophantineMin :: C a => a -> a -> a -> Maybe (a, a)

Like diophantine, but a is minimal with respect to the measure function of the Euclidean algorithm.

diophantineMulti :: C a => a -> [a] -> Maybe [a]

chineseRemainder :: C a => (a, a) -> (a, a) -> Maybe (a, a)

Not efficient enough, because GCD/LCM is computed twice.

chineseRemainderMulti :: C a => [(a, a)] -> Maybe (a, a)

For Just (b,n) = chineseRemainder [(a0,m0), (a1,m1), ..., (an,mn)] and all x with x = b mod n the congruences x=a0 mod m0, x=a1 mod m1, ..., x=an mod mn are fulfilled.

Properties

propMaximalDivisor :: C a => a -> a -> a -> Property

propGCDDiophantine :: (Eq a, C a) => a -> a -> Bool

propExtendedGCDMulti :: (Eq a, C a) => [a] -> Bool

propDiophantine :: (Eq a, C a) => a -> a -> a -> a -> Bool

propDiophantineMin :: (Eq a, C a) => a -> a -> a -> a -> Bool

propDiophantineMulti :: (Eq a, C a) => [(a, a)] -> Bool

propDiophantineMultiMin :: (Eq a, C a) => [(a, a)] -> Bool

propChineseRemainder :: (Eq a, C a) => a -> a -> [a] -> Property

propDivisibleGCD :: C a => a -> a -> Bool

propDivisibleLCM :: C a => a -> a -> Bool

propGCDIdentity :: (Eq a, C a) => a -> Bool

propGCDCommutative :: (Eq a, C a) => a -> a -> Bool

propGCDAssociative :: (Eq a, C a) => a -> a -> a -> Bool

propGCDHomogeneous :: (Eq a, C a) => a -> a -> a -> Bool

propGCD_LCM :: (Eq a, C a) => a -> a -> Bool

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