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 $n\ge N$, [itex]c_0= 1[/tex], $c_n= 0$ for n> 0.[/itex]