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When taking the completion a rational function field, say $k(t)$, with respect to the place at infinity, most books refer to this using the notation $k((1/t))$.

Since $k((t)) = k((1/t))$ (EDIT: this is not true. See Paul Garrett's answer below), why is this notation used? Is it just to remind us that the field was completed using the valuation at infinity, or perhaps to remind us that a prime element for the place at infinity is $1/t$? Are there reasons other than this?

Thanks!!

John

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It is not the case that $k((t))=k((1/t))$, because these half-infinite formal Laurent series are infinite in opposite senses. That is, the completion at $0$ gives $\sum_{n\ge -N} c_n\,t^n$, while the completion at infinity gives $\sum_{n\ge -N} c_n\,(1/t)^n$, which is $\sum_{n\le N} c_{-n}\,t^n$, for comparison.

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    Thanks! That was very helpful.2012-08-18