Wikipedia: ModularArithmetic

Difference (from prior major revision) (minor diff)

Changed: 1c1

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.

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.

Changed: 5,8c5,7

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

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

Changed: 10,13c9,11

* 0 1 2
0 0 0 0
1 0 1 2
2 0 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

Changed: 15c13

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.

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.