AlgebraicStructureTraits_::Gcd Concept Reference

Definition

AdaptableBinaryFunction providing the gcd.

The greatest common divisor ( \( gcd\)) of ring elements \( x\) and \( y\) is the unique ring element \( d\) (up to a unit) with the property that any common divisor of \( x\) and \( y\) also divides \( d\). (In other words: \( d\) is the greatest lower bound of \( x\) and \( y\) in the partial order of divisibility.) We demand the \( gcd\) to be unit-normal (i.e. have unit part 1).

\( gcd(0,0)\) is defined as \( 0\), since \( 0\) is the greatest element with respect to the partial order of divisibility. This is because an element \( a \in R\) is said to divide \( b \in R\), iff \( \exists r \in R\) such that \( a \cdot r = b\). Thus, \( 0\) is divided by every element of the Ring, in particular by itself.

Refines

AdaptableBinaryFunction

See also

AlgebraicStructureTraits

Types
typedef unspecified_type	result_type
	Is AlgebraicStructureTraits::Type.
typedef unspecified_type	first_argument
	Is AlgebraicStructureTraits::Type.
typedef unspecified_type	second_argument
	Is AlgebraicStructureTraits::Type.

Operations
result_type	operator() (first_argument_type x, second_argument_type y)
	returns \( gcd(x,y)\).
template<class NT1 , class NT2 >
result_type	operator() (NT1 x, NT2 y)
	This operator is defined if NT1 and NT2 are ExplicitInteroperable with coercion type AlgebraicStructureTraits::Type.