Mod(prime) is a field mod(non-prime) is not

[General_Sax]

Let n ≥ 2 be a natural number. Show that Z/Zn is a field if and only if n is a prime
number

Now, I can show that if n is prime then Z/Zn is a field

a = a
b = a^n-2

a*b = a^n-1 = 1 (mod n) --> Fermat's little theorem

However, I can't really seem to show that there is no multiplicative inverse for Z/Zn where n is not prime.

First question: a =/=b correct?

i know that there is the whole if gcd(a,n) = 1 then there is a multiplicative inverse, but I can't really see how to leverage this fact.

Any help would be much appreciated.

 

[General_Sax]

i know that there is the whole if gcd(a,n) = 1 then there is a multiplicative inverse, but I can't really see how to leverage this fact.

Not only that, but I don't truly understand why this is.

 

[Deveno]

for non-prime n,

what you want to do is find a,b ≠ 0 in Z/Zn such that:

ab = 0 (mod n).

these elements cannot be invertible.

suppose 1/a existed. then:

(1/a)(ab) = [(1/a)(a)]b = 1b = b ≠ 0 (mod n)

BUT...

(1/a)(ab) = (1/a)0 = 0 (mod n), a contradiction.

(1/b) can be shown not to exist in a similar fashion (multiply on the right).

since n is not prime, there is some 1 < d < n with d|n.

so n = dk, where 1 < d,k < n.

thus d,k ≠ 0 (mod n) but dk = n = 0 (mod n).

 

[General_Sax]

Thanks

 

[blue_raver22]

I understand your struggle with this problem. The statement that Mod(prime) is a field while Mod(non-prime) is not is a well-known fact in mathematics. In order to prove this, we need to first understand the definition of a field.

A field is a mathematical structure that consists of a set of elements, along with two operations, addition and multiplication, that follow specific rules. These rules include closure, associativity, commutativity, existence of identity elements, and existence of inverse elements.

Now, let's consider Z/Zn, the set of integers modulo n. This set consists of the remainders when dividing any integer by n. For example, in Z/Z6, we have the elements {0, 1, 2, 3, 4, 5}.

If n is a prime number, then Z/Zn is a field. This is because for any element a in Z/Zn, we can find its inverse element b such that a*b = 1 (mod n). This is due to the fact that in a prime field, every element has a multiplicative inverse. This can be seen in your example, where a = an-2 is the inverse of itself.

However, if n is not a prime number, then Z/Zn is not a field. This is because there exist elements in Z/Zn that do not have a multiplicative inverse. This is due to the fact that in a non-prime field, not every element has a multiplicative inverse. In other words, there exists an element a in Z/Zn such that gcd(a,n) ≠ 1, which means that there is no integer b that satisfies the equation a*b = 1 (mod n). This can be seen in your example, where n = 6 and a = 2. In this case, gcd(2,6) = 2, and there is no integer b that satisfies the equation 2*b = 1 (mod 6).

In conclusion, we can say that Z/Zn is a field if and only if n is a prime number. This is because only in a prime field, every element has a multiplicative inverse, which is a necessary condition for a field to exist. Therefore, the statement "Mod(prime) is a field while Mod(non-prime) is not" is true and can be proven using the definition of a field