Field of formal Laurent series

Remarks

The inner sum $\sum_{i + j = n} a_i b_j$ is finite: if $a_i = 0$ for $i < N$ and $b_j = 0$ for $j < M$, only the pairs with $N \le i \le n - M$ contribute. Example: $f = X - X^2$ has the inverse $g = \sum_{n \ge -1} X^n = X^{-1} + 1 + X + \cdots$; indeed $f g = (1 - X)(1 + X + X^2 + \cdots) = 1$. Non-example: the family with $a_n = 1$ for all $n \in \Z$ is not a formal Laurent series, since no $N$ bounds its negative powers. Its square would need the infinite sum $\sum_{i + j = 0} 1$. The map $v(f) = \min \set{n : a_n \ne 0}$ for $f \ne 0$ is a discrete valuation on $k((X))$ with uniformizer $X$, and $k((X))$ is complete for it. Its valuation ring is the ring $k[\![X]\!]$ of formal power series, and $k((X))$ is the field of fractions of $k[\![X]\!]$.