# A canonical injection

1. Feb 8, 2008

### quasar987

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))

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

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....

2. Feb 8, 2008

### morphism

Which obvious ones did you think of?

3. Feb 8, 2008

### quasar987

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

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

4. Feb 8, 2008

### HallsofIvy

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