Is S Isomorphic to Any Commonly Known Fields?

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Homework Statement


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.

Homework Equations




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



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
 
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How about commonly known extension fields? Like the rationals extended by sqrt(3)?
 
I am not sure what that means
 
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.
 
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got it, thanks
 
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