³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980240446³2ip³2129.97.182.28³2data³2newauthor³31³3minor³30³3text³3The 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.
³3summary³3*³2tscreate³2980240446³2keepts³2980240489³2id³21076³2host³2cnts2p28.uwaterloo.ca³2revision³21³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980240489³2ip³2129.97.182.28³2data³2text³3The 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
| 0  1  2
| 0  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.
³3minor³30³3newauthor³30³3summary³3*³2tscreate³2980240446³2keepts³2980240507³2id³21076³2host³2cnts2p28.uwaterloo.ca³2revision³22³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980240507³2ip³2129.97.182.28³2data³2newauthor³30³3minor³31³3text³3The 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
| 0  0  1  2  
| 0  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.
³3summary³3*³2tscreate³2980240446³2keepts³2980240599³2id³21076³2host³2cnts2p28.uwaterloo.ca³2revision³23³1username³2JoshuaGrosse³2name³2text_default³2version³21³2ts³2980240599³2ip³2129.97.182.28³2data³2text³3The 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.
³3minor³31³3newauthor³30³3summary³3*³2tscreate³2980240446³2keepts³2980697739³2id³21076³2host³2cnts2p28.uwaterloo.ca³2revision³24