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Formal power series

DEFINITION 01.1   Let $ A$ be a commutative ring. Let $ X$ be a variable. A formal power series in $ X$ over $ A$ is a formal sum

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$\displaystyle \sum_{i=0}^\infty a_i X^i \quad ( a_i \in A )
$

We denote by $ A[[X]]$ the ring of formal power series in $ X$ . Namely,

$\displaystyle A[[X]]=\{ \sum_{i=0}^\infty a_i X^i; a_i \in A\}.
$

For any element $ f=\sum_n a_n X^n$ of $ A[[X]]$ , we define its order as follows:

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$\displaystyle {\mathrm{ord}}(f)=\inf\{n; a_n \neq 0\}.
$

Then we may define a metric on $ A[[X]]$ .

$\displaystyle d(f,g)=\frac{1}{2^{{\mathrm{ord}}(f-g)}}
$

EXERCISE 01.1   Show that $ (A[[X]],d)$ is a complete metric space.

EXERCISE 01.2   Show that $ A[[X]]$ is a topological ring. That means, it is a topological space equipped with a ring structure and operations (the addition and the multiplication) is continuous.



2013-04-05