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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.

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Last edited January 29, 2001 12:02 am by JoshuaGrosse (diff)