Z/3z

I need to construct the ring Z/3Z by,

1- Identifying the ring 3Z
2- Defining Z/3Z using an equivalence relation
3- Identifting all of the equivalence classes in Z/3Z
4- Constructing tables for addition and multiplication in Z/3Z.

Would Z/3Z be a field?

Reply 1

JamesF

Ok, firstly do you know what the ring of 3Z is? Do you know what an equivalence relation is?

Reply 2

JamesF

The ring 3Z is every integer multiple of 3, now to define an equivalence relation.
You need to prove 3 properties,
1) a~a
2) a~b => b~a
3) a~b , b~c => a~c

If you define your relation (~) as a~b means that 3 divides a-b, then you can check the properties hold, and that this relation partitions the ring Z into 3 sets.

The addition and multiplication tables should be simple enough to write out, then you can use them to find out whether it is a field or not.

Reply 3

JFN

Thanks, I knew that. I was actually trying to find the necessary condition for a natural number n such that Z/nZ is a field rather than a ring - I think I got it.

Reply 4

mathz

n prime. a quick reason for this is if it is not prime then it is composite say n=kl with k,l<n if we pick equivalence classes to be [0],[1],[2],...[n-1] therefore in Z/nZ we have [k][l]=[n] and so is in [0] so have a zero divisor (that is 2 non-zero numbers multiplied to give 0) which does not happen in a field
more generally for p prime it can shown pZ is a maximal ideal and it can be shown for any ring R, maximal ideal I, that R/I is a field.