5.20 EuclideanRemainder

EuclideanRemainder( r, m )
EuclideanRemainder( R, r, m )

In the first form EuclideanRemainder returns the remainder of the ring element r modulo the ring element m in their default ring. In the second form EuclideanRemainder returns the remainder of the ring element r modulo the ring element m in the ring R. The ring R must be a Euclidean ring (see IsEuclideanRing) otherwise an error is signalled.

A ring R is called a Euclidean ring, if it is an integral ring, and there exists a function delta, called the Euclidean degree, from R-{0_R} to the nonnegative integers, such that for every pair r in R and s in R-{0_R} there exists an element q such that either r - q s = 0_R or delta(r - q s) < delta( s ). The existence of this division with remainder implies that the Euclidean algorithm can be applied to compute a greatest common divisors of two elements, which in turn implies that R is a unique factorization ring. EuclideanRemainder returns this remainder r - q s.

    gap> EuclideanRemainder( 16, 3 );          
    1
    gap> EuclideanRemainder( Integers, 201, 11 );
    3 

EuclideanRemainder calls R.operations.EuclideanRemainder( R, r, m ) in order to compute the remainder and returns the value.

The default function called this way uses QuotientRemainder in order to compute the remainder.

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