Proposition 12

— Every non-zero element $u$ of $K((X))$ may be written in a unique way as $u = X^k v$, where $k \in \mathbf{Z}$ and $v$ is a formal power series in $X$ of order 0.

Let $u = w/t$, where $w, t$ are non-zero elements of $K[[X]]$. We have $w = X^r w_1, t = X^s t_1$, where $r, s \in \mathbf{N}$ and $w_1, t_1$ are formal power series of order 0, hence invertible in $K[[X]]$ (IV, p. 30, Prop. 6). Then $u = X^{r-s} w_1 t_1^{-1}$ and $w_1 t_1^{-1}$ is a formal power series of order 0.

Let us prove the uniqueness. Suppose that $u = X^{k_1} v_1 = X^{k_2} v_2$ where $k_1, k_2 \in \mathbf{Z}$ and $v_1, v_2$ are formal power series of order 0. Since $X^{k_1 - k_2} = v_2 v_1^{-1}$ is a formal power series of order 0, we have $k_1 = k_2$ whence $v_1 = v_2$ and this proves the uniqueness assertion.

We shall say that the elements of $K((X))$ are generalized formal power series in $X$ with coefficients in $K$, or simply formal power series when no confusion can arise (the elements of $K[[X]]$ are then called formal power series with positive exponents); if $u \neq 0$, the integer $k$ defined in Prop. 12 is also called the order of $u$ and is written $\omega(u)$, even if it is $< 0$; we also put $\omega(0) = \infty$. It may be verified at once that

still hold for generalized formal power series. In particular, if $u \neq 0$, then $w(u^{-1}) = -w(u)$. \* In other words (Comm. Alg., VI, § 3, No. 6, p. 392, Def. 3), w is a normalized discrete valuation of the field $K((X))$. \*

For each integer $n \in \mathbf{Z}$ let $p_n$ be the set of all $u \in K((X))$ such that $\omega(u) \geq n$. Then (p.), , , is a decreasing sequence of subgroups of the additive group $K((X))$, with intersection 0 ; there exists thus a topology on $K((X))$, invariant under translation, for which $(p_n)_{n \in \mathbf{Z}}$ is a fundamental system of neighbourhoods of 0 (Gen. Top., III, p. 223). We can easily verify that $K((X))$ is a topological field (Gen. Top., III, p. 281) and that $K[[X]]$ is an open and closed subspace of $K((X))$.

Let $(\alpha_n)_{n \in \mathbf{Z}}$ be a family of elements of $K$, and suppose that there exists an integer N such that $a_n = 0$ for all $n < N$. Then the family' $(\alpha_n X^n)_{n \in \mathbf{Z}}$ is summable in $K((X))$ (Gen. Top., III, p. 263, Cor.) ; put $u = \sum_{n \in \mathbf{Z}} \alpha_n X^n$, then $u = 0$ if and only if $a_n = 0$ for all $n$; otherwise the order of $u$ is the least integer k such that $\alpha_k \neq 0$. Finally every element of $K((X))$ may be written in a unique fashion in the form $\sum_{n \in \mathbf{Z}} \alpha_n X^n$, where the sequence $(\alpha_n)$ satisfies $\alpha_{-n} = 0$ for all sufficiently large n.

Since the ring $K[X]$ is a subring of $K[[X]]$, every rational fraction $u/v \in K(X)$ (u, v being polynomials in X) may be identified with the (generalized) formal power series $uv^{-1}$ of $K((X))$, which we shall call its expansion at the origin ; the field $K(X)$ is thus identified with a subfield of $K((X))$.