Wikipedia 10K Redux by Reagle from Starling archive. Bugs abound!!!

<-- Previous | Newer --> | Current: 980240446 JoshuaGrosse at Tue, 23 Jan 2001 09:00:46 +0000.

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.