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作者  new1111111 (new) 站內  ALGEBRA
標題  Re: [問題] field
時間  2009/07/22 Wed 18:08:21

※ 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~ = =+ )


有錯請指正,謝謝!

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

Let a belongs to F. We consider a surjective ring homomorphism

f: F ----> F, f(r)=r^p. Thus a = x^p for some x in F.

∵ x belongs to F, x = y + z for some y and z belong to F.

============>  a = (y + z)^p = y^p + z^p.

If p = 2, the proof is completed. Now suppose p is an odd prime.

Let b = y^m, c = z^m where m =(1/2)*[p^(n+1) - p]

============> b^2 = y^p and c^2 = z^p

============> a = b^2 +c^2.
                            □

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□ Modify: 2009/07/22 Wed 21:32:39  math204.math.ncku.edu.tw 修改

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