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+0=0    1+0=1    2+0=2
  0+1=1    1+1=2    2+1=0
  0+2=2    1+2=0    2+2=1

  0*0=0    1*0=0    2*0=0
  0*1=0    1*1=1    2*1=2
  0*2=0    1*2=2    2*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.