### 博碩士論文 90221001 完整後設資料紀錄

 DC 欄位 值 語言 DC.contributor 數學系 zh_TW DC.creator 施柏如 zh_TW DC.creator Po-Ju Shih en_US dc.date.accessioned 2004-1-16T07:39:07Z dc.date.available 2004-1-16T07:39:07Z dc.date.issued 2004 dc.identifier.uri http://ir.lib.ncu.edu.tw:88/thesis/view_etd.asp?URN=90221001 dc.contributor.department 數學系 zh_TW DC.description 國立中央大學 zh_TW DC.description National Central University en_US dc.description.abstract 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. en_US DC.subject Legendre's Theorem en_US DC.title Legendre的定理在Z[i]和Z[w]的情形 zh_TW dc.language.iso zh-TW zh-TW DC.title Legendre's Theorem in Z[i] and in Z[w] en_US DC.type 博碩士論文 zh_TW DC.type thesis en_US DC.publisher National Central University en_US