ModularArithmetic

The ModularArithmetics are the images of the IntegerNumbers under group/ring HomoMorphisms. Such an operation is going to zero out some NormalSubgroup/Ideal, and these turn out to be precisely the sets of the form pZ for some integer p; the resulting group/ring is denoted Zp.

To put it another way, Zp consists of the remainders {0,1,...,p-1}, so that p=0. For instance, Z3 has the following addition and multiplication tables:

+ 0 1 2

0 0 1 2

1 1 2 0

2 2 0 1

• 0 1 2

0 0 0 0

1 0 1 2

2 0 2 1

When p is a composite number, the factors of p are going to turn out to be ZeroDivisors. When p is prime, these don't exist, and so Zp is an IntegralDomain and in fact necessarily a field.