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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Euler's criterion ： ウィキペディア英語版 Euler's criterion In number theory Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely, Let ''p'' be an odd prime and ''a'' an integer coprime to ''p''. Then〔Gauss, DA, Art. 106〕 :$$ a^} \equiv \begin \;\;\,1\pmod& \textx \texta\equiv x^2 \pmod\\ -1\pmod& \text \end Euler's criterion can be concisely reformulated using the Legendre symbol:〔Hardy & Wright, thm. 83〕 :$$ \left(\frac\right) \equiv a^ \pmod p. The criterion first appeared in a 1748 paper by Euler.〔Lemmermeyer, p. 4 cites two papers, E134 and E262 in the Euler Archive〕 ==Proof== The proof uses the fact that the residue classes modulo a prime number are a field. See the article prime field for more details. The fact that there are quadratic residues and the same number of nonresidues (mod ) is proved in the article quadratic residue. Fermat's little theorem says that :$$ a^\equiv 1 \pmod p (Assume throughout this solution that is not 0 mod ). This can be written as :$$ \left( a^}-1 \right)\left( a^}+1 \right) \equiv 0 \pmod p. Since the integers mod form a field, one or the other of these factors must be congruent to zero. Now if is a quadratic residue, , :$$ a^}\equiv ^} \equiv x^\equiv1\pmod p. So every quadratic residue (mod ) makes the first factor zero. Lagrange's theorem says that there can be no more than values of that make the first factor zero. But it is known that there are distinct quadratic residues (mod ) (besides 0). Therefore they are precisely the residue classes that make the first factor zero. The other residue classes, the nonresidues, must be the ones making the second factor zero. This is Euler's criterion. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Euler's criterion」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.