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?
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?
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.
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...
Right. Got it.
The definition I was given does not require associativity to be a ring.
See: Wikipedia: Nonassociative ring
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...