# Finding that S is a Field

1. Dec 12, 2009

### bokasabi

1. The problem statement, all variables and given/known data
let S be the ring of all pairs (q,s) with q,s being rational.
define (q,s) + (q',s') = (q+q',s+s')
and (q,s)(q',s') = (qs'+q's,3qq'+ss')
Prove that S is a field.

2. Relevant equations

Try proving that it is isomorphic to something else using the First homomorphism theorem for rings.

3. The attempt at a solution

I can not find a homomorphism between S and any commonly known fields like complex numbers, real numbers, rational numbers, etc

2. Dec 12, 2009

### Dick

How about commonly known extension fields? Like the rationals extended by sqrt(3)?

3. Dec 12, 2009

### bokasabi

I am not sure what that means

4. Dec 12, 2009

### Dick

The set of numbers of the form p+q*sqrt(3) where p and q are rational. It's a field. Can you prove it? It has dimension two as a vector space over the rationals. I think of that as pretty commonly known.

Last edited: Dec 12, 2009
5. Dec 12, 2009

### bokasabi

got it, thanks