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Prime factorization of rationals

  1. Jul 6, 2008 #1
    It occurred to me that the rationals Q have also a unique prime factorization, as long as you allow negative exponents on the factorization.

    If a/b is a rational, then both a and b have a unique (integer) prime factorization, and the fraction can be expressed uniquely as a product of primes, raised to the difference of the exponents found in the respective prime factors of a and b. Note that a/b does not even need to be reduced for this to work.

    I find this a beautiful idea, but I ignore how to use it further, or what else can be constructed using it.

    Edit: oops, except for zero... bye-bye to groups, rings, fields...
     
  2. jcsd
  3. Jul 6, 2008 #2

    matt grime

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    Good news: this idea is useful. What you've done is come very close to discovering non-archimedian norms, which leads to the study of p-adic anlysis.

    Bad news: you can't call it prime factorisation, since everything is a unit.
     
  4. Jul 6, 2008 #3
    Thanks, Matt. What do you mean, everything is a unit?
     
  5. Jul 6, 2008 #4

    CRGreathouse

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    Evey nonzero rational has a reciprocal that is a nonzero rational. In general, a 'unit' is a member of a ring which has a multiplicative inverse in the ring.
     
  6. Jul 6, 2008 #5

    matt grime

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    And (unique) prime factorisations, when they exist, are only ever defined 'up to units'. It is just about not mixing and matching your definitions, that's all.
     
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