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A canonical injection

  1. Feb 8, 2008 #1

    quasar987

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    1. The problem statement, all variables and given/known data
    Given a field F, I'm trying to find an injection from the set of formal Laurence series F((x))

    [tex]\sum_{n\geq N}^{+\infty}a_nx^n, \ \ \ \ \ N\in\mathbb{Z}[/tex]

    to the ring of fractions of formal power series [tex]\mathbb{Q}(F[[x]])[/tex]

    [tex]\frac{\sum_{n=0}^{+\infty}a_nx^n}{\sum_{n=0}^{+\infty}b_nx^n}[/tex]

    (where the denominator is not a divisor of 0 in F[[x]])


    I've tried all the obvious mapping I could think of, but they failed to be injections....
     
  2. jcsd
  3. Feb 8, 2008 #2

    morphism

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    Which obvious ones did you think of?
     
  4. Feb 8, 2008 #3

    quasar987

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    For instance, truncate the part of the series when n is negative.

    Or send the part where n is negative on the denumenator.
     
  5. Feb 8, 2008 #4

    HallsofIvy

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    One I would consider extremely obvious would be to map
    [tex]\sum_{n\geq N}^{+\infty}a_nx^n[/tex]
    to
    [tex]\frac{\sum_{n=0}^{+\infty}b_nx^n}{\sum_{n=0}^{+\infty}c_nx^n}[/tex]
    where [itex]b_n= 0[/itex] if n< N, [itex]b_n= a_n[/itex] if [itex]n\ge N[/itex], [itex]c_0= 1[/tex], [itex]c_n= 0[/itex] for n> 0.
     
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