QuotientRemainder( r, m )
QuotientRemainder( R, r, m )
In the first form QuotientRemainder
returns the Euclidean quotient and
the Euclidean remainder of the ring elements r and m in their default
ring as pair of ring elements. In the second form QuotientRemainder
returns the Euclidean quotient and the Euclidean remainder of the ring
elements r and 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.
QuotientRemainder
returns this quotient q and the remainder r - q
s.
gap> qr := QuotientRemainder( 16, 3 ); [ 5, 1 ] gap> 3 * qr[1] + qr[2]; 16 gap> QuotientRemainder( Integers, 201, 11 ); [ 18, 3 ]
QuotientRemainder
calls R.operations.QuotientRemainder( R, r,
m )
and returns the value.
The default function called this way is RingOps.QuotientRemainder
,
which just signals an error, because there is no default function to
compute the Euclidean quotient or remainder of one ring element modulo
another. Thus Euclidean rings must overlay this default function with
other functions.
GAP 3.4.4