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Homework Help: Finding that S is a Field

  1. Dec 12, 2009 #1
    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. jcsd
  3. Dec 12, 2009 #2


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    How about commonly known extension fields? Like the rationals extended by sqrt(3)?
  4. Dec 12, 2009 #3
    I am not sure what that means
  5. Dec 12, 2009 #4


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    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
  6. Dec 12, 2009 #5
    got it, thanks
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