Dialogue on n-Color Arithmetic (3)

Part 3:
The n-Number Arithmetic

Si Emo was astonished with his finding that small finite set of numbers has an arithmetic structure similar to the arithmetic of the infinitely large set of real numbers. But his grandpa Ki Algo enlightened his mind, by showing that such 2-number and 3-number fields is similar to modulo 2 and modulo 3 arithmetic of numbers. Let us listen to the continuing dialogue on n-Number Arithmetic.
Si Emo: Grandpa, I have visited the 4-Number Island and found out nothing interesting in it. It is populated by numbers 0, 1, 2 and 3. Its addition and multiplication rules is nothing but the rules for modulo 4 arithmetic. The table for addition is

+	0	1	2	3
0	0	1	2	3
1	1	2	3	0
2	2	3	0	1
3	3	0	1	2

and the table for multiplication is

.	0	1	2	3
0	0	0	0	0
1	0	1	2	3
2	0	2	1	2
3	0	3	2	1

Ki Algo:  What did you find in 5-number island?
Si Emo: Si Emo:  Nothing interesting in the island. It is populated by numbers 0, 1, 2, 3 and 4. And its arithmetic is similar to modulo 5 arithmetic.

n-Number Arithmetic

Ki Algo: Now, it seems that we can generalized this arithmetic of n numbers: 0, 1, 2, ....and n-1.
The system also defined by two operation: addition + and multiplication . 

a + b = the remainder of (a+b) when it is divided by n
a . b = the remainder of (a.b) when it is divided by  n
Si Emo: Is the arithmetic of n-number, for all natural number n, similar to real number arithmetic for all natural number n.
Ki Algo:  No!
Si Emo:  Why?
Ki Algo:  If  the number n is a nonprime number, then there is some strange property emerge for n is nonprime number which is not equal to powers of prime number
Si Emo:  Let me test it. See, 6 is non prime number. It is 3 times 2. For 6-numbers 2 and 3, they are multiplied to 0. Yes. It is so strange because, in the field of real numbers, if two numbers is multiplied to 0, then one of the two numbers must be 0.
Ki Algo: Mathematicians called 2 and 3 as Zero Divisors for 6-numbers. The existence of  zero divisors makes the arithmetic of 6-number not a field anymore. It's arithmetic is a Ring. Some mathematician call the Field as Division Ring.
Si Emo:  So the 10-number arithmetic is also not a Field, because 10=2.5, but 7-number arithmetic is a field like the 2-number, 3-number and 5-number arithmetic.
Ki Algo:  Yes. All p-number arithmetic, call it F_p, is a field if p is a prime number.
Si Emo:  OK. I know that 4 is not a prime number, but F₄ is a field
Ki Algo: Well! That's another matter. F_n is a field if n is a k-th power of a prime number p or n=p^k to prove that it is really a field is a tricky business, but it is similar in structure to the arithmetic of the  many-colored  numbers that live in the islands, in the lagoons of the Numberland, that Si Nessa visited yesterday. Let us listen to her story to Ni Suiti later.