Let

Prove that both and are fields but that they are not isomorphic.

Note: rings are commutative and associative with unity.

To prove they are fields, just find the general multiplicative inverse.

How do I prove they are not isomorphic?

- Mar 29th 2011, 03:01 PMjoestevensProve that both rings are fields but that they are not isomorphic.
Let

Prove that both and are fields but that they are not isomorphic.

Note: rings are commutative and associative with unity.

To prove they are fields, just find the general multiplicative inverse.

How do I prove they are not isomorphic? - Mar 29th 2011, 04:29 PMhatsoff
You should have a theorem available to show that and . Another theorem says that is a field if is irreducible over . So just show that are irreducible (using Eisenstein). It follows that both are fields.

Next show that any nonzero homomorphism has . It follows that , which means . But if , then . Use this to show that is rational, a contradiction. So there are no nonzero homomorphism, i.e. no isomorphisms. - Mar 29th 2011, 06:47 PMjoestevens
- Mar 29th 2011, 06:50 PMTheChaz
- Mar 30th 2011, 02:28 AMSwlabr
- Mar 30th 2011, 07:49 AMjoestevens
For my current problem set, all rings are considered associative and commutative with unity.

I agree that his was is neater.... - Mar 30th 2011, 08:04 AMSwlabr
The fact that they are assumed it is beside the point. I mean, you should be able to spot right off that is commutative with unity! To attack the problem properly, you should say that this is so. It doesn't require proof, just a line...

Also, all rings are associative... - Mar 30th 2011, 08:10 AMjoestevens
Right. Got it.

The definition I was given does not require associativity to be a ring.

See: Wikipedia: Nonassociative ring - Mar 30th 2011, 08:28 AMSwlabr
That is...odd. The only `real' benefit of this for an undergrad which I can see is that Lie algebras are nonassociative rings, so you can slip in a section about this. However, both Lie algebras and (associative) rings are rich enough areas that rolling them into one is a bit over the top...