The ring Z/mZ consists of representatives for the residue classes of integers modulo m > 1. This online help node and the nodes below it describe the operations in Magma for such rings and their elements.

Sometimes it is convenient to perform certain modular operations on integers without explicitly creating the appropriate residue class rings -- the possibilities for doing so are described at the end of the section on the Ring of Integers.

Subsections

• Representation
• Coercion
• Homomorphisms

At any stage during a session, Magma will have at most one copy of Z/mZ present, for any m>1; that is, different names for the same residue class ring will n facd be different references to copies of the same structure. This saves memory space, and avoids confusion about different but isomorphic structures.

If m is a prime number, the ring Z/mZ forms a field; however, Magma has special functions for dealing with finite fields. The operations described here should not be used for finite field calculations: the implementation of finite field arithmetic in Magma takes full advantage of the special structure of finite fields and leads to superior performance.

Coercion

Automatic coercion takes place between Z/mZ and Z so that a binary operation like + applied to an element of Z/mZ and an integer will result in a residue class from Z/mZ.

Using !, elements from a prime field GF(p) can be coerced into Z/pZ, and elements from Z/pZ can be coerced into GF(p^r). Also, transitions between Z/mZ and Z/nZ can be done with ! provided that m divides n or n divides m. In cases where there is a choice -- such as when an element r from Z/mZ is coerced into Z/nZ with m dividing n -- the result will be the residue class containing the representative for r.

> r := ResidueClassRing(3) ! 5;
> r;
2
> ResidueClassRing(6) ! r;
2

Ring homomorphisms with domain Z/mZ are completely determined by the image of 1. As usual, we require our homomorphisms to map 1 to 1. Therefore, the general homomorphism constructor with domain Z/mZ takes no arguments.

hom< R -> S | > : RngIntRes, Rng -> Map

Given a residue class ring R, and a ring S, create a homomorphism from R to S, determined by f(1_R) = 1_S. Note that it is the responsibility of the user that the map defines a homomorphism!