Is There a Canonical Injection from F((x)) to Q(F[[x]])?

[quasar987]

Homework Statement

Given a field F, I'm trying to find an injection from the set of formal Laurence series F((x))

\sum_{n\geq N}^{+\infty}a_nx^n, \ \ \ \ \ N\in\mathbb{Z}

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

\frac{\sum_{n=0}^{+\infty}a_nx^n}{\sum_{n=0}^{+\infty}b_nx^n}

(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...

 

[morphism]

Which obvious ones did you think of?

 

[quasar987]

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

Or send the part where n is negative on the denumenator.

 

[HallsofIvy]

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