power series field

《Algebraic Number Theory》 · 第 127 页 · Chapter II，Section 4「Completions」，定义段

In the case of the rational function field $k(t)$ and the valuation $v_{\mathfrak p}$ attached to a prime ideal $\mathfrak p=(t-a)$ of $k[t]$, we may take as a system of representatives the field of coefficients $k$ itself. The completion then turns out to be the field of formal power series $k((x))$, $x=t-a$, consisting of all formal Laurent series $f(t)=(t-a)^m(a_0+a_1(t-a)+a_2(t-a)^2+\cdots)$, with $a_i\in k$ and $m\in\mathbb Z$.