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Well, it is hard to find a good references on The field of Laurent series on finite fields. Let $F_q$ be any finite field, and denote $F_q((t))$ is the field of Laurent series on $F_q$. Please show me, how to define it, operations, absolute value and prove that its absolute value is non-Archimedian.

Please give me useful links or references. Thanks ~

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Have you seen Formal Laurent series? – lhf May 14 '11 at 2:32
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@lhf: yes I have, but I need more informations than that. – Jie Fan May 14 '11 at 2:40
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What are you having trouble with? – Qiaochu Yuan May 14 '11 at 2:55
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We can define the valuation and this valuation is non-Archimedian . Specially Laurent series are local field. What are you looking for? – Babak Miraftab Jun 7 '12 at 8:06
  1. $F_q((x)):=\left\{\sum_{n=k}^{\infty}a_nz^n:\ \text{for } n\in\mathbb{N},\text{ and }a_n\in F_q,\ n=k,k+1,\ldots\right\}$

  2. For $F:=\sum a_nz^n, G:=\sum b_nz^n\in F_q((z))$ and $a\in F_q$ we have

    1. $F+G:=\sum (a_n+b_n)z^n$
    2. $aF:=\sum aa_nz^n$
    3. $FG:=\sum c_nz^n$, with $c_n:=\sum_k a_nb_{n-k}$
  3. For $F:=\sum_{n=k}^{\infty}a_nz^n$, with $a_k\neq0$ we define $|F|:=q^{-k}$

The absolute value of $3$ is non-Archimedean. In fact, in general, we have

$$|F+G|\leq\max\left(|F|,|G|\right).$$

In particular $|z|=q^{-1}$ and therefore $|n\cdot z|=|z+z+\ldots+z|\leq\max(|z|,\ldots,|z|)=q^{-1}$. On the other hand we have $|1|=q^0=1$. Therefore we can never make $|n\cdot z|>1$ for any $n\in\mathbb{N}$.

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