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作者  new1111111 (new) 站內  ALGEBRA
標題  Re: [問題] field
時間  2009/07/23 Thu 23:07:21

※ 引述《george (...)》之銘言:
>  Every element in a finite field can be written as a sum of two squares.
>  ( 一樣夢幣100萬 呵呵~ = =+ )


不好意思  我之前po的證錯了 我後來又想了一個

請各位高手幫我看一下 謝謝!

Let F be a finite field with char F = p and │F│= p^n.

====> F\{0} is cyclic and  F\{0} = < r > for some r belongs to F.

Suppose a belongs to F and a can not be written as a sum of two squares.

====>                 a = r^2 + r^(k_1)

                        = r^4 + r^(k_2)

                        = r^6 + r^(k_3)
                        .
                        .
                        .
                        = r^(p^n - 1) + r^(k_m)

where m = (1/2)*(p^n - 1) and each k_i is odd.

====>                 ma = the sum of elements in F = 0.

∵ m is not a multiple of p ==> a = 0, a contradiction.

                                                       □


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