Legendre的定理在Z[i]和Z[w]的情形;Legendre's Theorem in Z[i] and in Z[w]

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题名:	Legendre的定理在Z[i]和Z[w]的情形;Legendre's Theorem in Z[i] and in Z[w]
作者:	施柏如;Shih,Po-Ju
贡献者:	數學研究所
关键词:	Legendre's Theorem
日期:	2004-01-16
上传时间:	2014-05-08 15:26:30 (UTC+8)
出版者:	國立中央大學
摘要:	none ;This thesis studies the Diophantine equation egin {eqnarray} ax^{2}+by^{2}+cz^{2}=0, end {eqnarray} which was investigated by Legendre when the coefficients are rational integers. Without loss of generality, we may assume that $a,b,c$ are nonzero integers, square free, and pairwise relatively prime. Legendre proved that the equation $ax^{2}+by^{2}+cz^{2}=0$ has a nontrivial integral solution if and only if egin{itemize} item[ m (i)] $a, b, c$ are not of the same sign, and item[ m(ii)] $-bc, -ac,$ and $-ab$ are quadratic residues of $a,b,$ and $c$ respectively. end{itemize} The purpose of this thesis is to extend Legendre's Theorem by carrying over the cases with the coefficients and unknowns in ${mathbb Z}[i]$ and in ${mathbb Z}[omega]$, where $i$ is a square root of $-1$ and $omega$ is a cubic root of unity. More precisely, we show that the necessary and sufficient conditions for the Diophantine equation $ax^{2}+by^{2}+cz^{2}=0$ having a nontrivial solution over ${mathbb Z}[i]$ is that $bc, ca,ab$ are quadratic residues mod $a,b,c$ respectively, and the equation having a nontrivial solution over ${mathbb Z}[omega]$ is that $-bc, -ca, -ab$ are quadratic residues mod $a,b,c$ respectively.
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