Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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 ~

share|improve this question
1  
Have you seen Formal Laurent series? –  lhf May 14 '11 at 2:32
    
@lhf: yes I have, but I need more informations than that. –  Hai Minh May 14 '11 at 2:40
8  
What are you having trouble with? –  Qiaochu Yuan May 14 '11 at 2:55
    
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 Answer 1

  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 $n\in\mathbb{N}$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.