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作者  krull (krull) 站內  ALGEBRA
標題  Re: [問題] field
時間  2009/07/24 Fri 20:03:58

※ 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~ = =+ )

F:= F_q : a finite field, q = p^n,  p : odd prime

let a in F

if a = 0, ok

if a is a square, ok

if a is not a square,

exist b in F* s.t b is not a square but b-1 is a square

(a result of basic number theory)

b^(-1)*a is a square ( a step not very sure = =, if q=p is ok )

=> X^2 = b^(-1)*a for some X in F*

=> a = b*X^2 = X^2 + (b-1)*X^2 = X^2 + (c*X)^2 for some c in F*


eg: p = 17

    1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16
   __ __    __          __ __              __      __  __  (底線為square)

let b = 3 , b^(-1) = 6

    3*X^2 = X^2 + 2*X^2 = X^2 + (6*X)^2 = 3 or 5 or 6 or 7 or 10 or 11 or 12

                                          or 14

 => X^2 = 1 or 13 or 2 or 8 or 9 or 15 or 4 or 16

 =>   X = 1 or 8  or 6 or 5 or 3 or 7  or 2 or 4

--

期待看到一個幾何的看法, or any other method of math, 哈哈~~

--
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  •FROM [krull 從 mcl240.mcl.math.ncu.edu.tw 發表]
□ Modify: 2009/07/24 Fri 22:13:43  mcl240.mcl.math.ncu.edu.tw 修改

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