### group, ring, field

**1. Closure:**when we operate two elements from the group, the result is also en element of the group.

**2. Associativity:**(a*b)*c = a*(b*c).

**3. Identity element:**there exists an indentity element e such that for any element a of the group, a*e = e*a = a.

**4. Inverse element:**for each element a of the group, there exists an element a

^{-1}in the group such that a*a

^{-1}= e.

_{p}, which refers to the integers modulo p, is also a group under the addition.

_{n}. Imagine we are in Z

_{6}, which means that since we are working modulo 6 we only have the integers 0, 1, 2, 3, 4, 5. 0 is not a generator. 1 is a generator. 2 is not a generator because 2+2 = 4 and 2+2+2 = 6 = 0, so we will never reach 1, 3 or 5. The same thing happens with 3 and 4, but 5 is a generator. Maybe you can see that the generators of Z

_{n}are the numbers that are coprime to n (see "Totient function"). An interesting remark is that if n is a prime number, then all the numbers in the group are coprime to n. Also note that you can imagine all the elements of Z

_{p}placed in the unit circle. If you put the pencil in one of the elements and add it to itself, you will go to another element (axiom of closure). If this first element that you chose is a generator, you will draw a star going through all the elements. Otherwise you will not. Here you can see an example with Z

_{6}:

_{p}. Also, p-adic numbers form a field, and why they are not a ring is because they accept inverses. Therefore, we have addition, substraction, multiplication and division. They are referred to as Q

_{p}. So if we have a p in the denominator of a fraction we are in Q

_{p}.

Note: in other sections we refer to the generators of the additive group modulo n as "generators".