Legendre's Theorem in Z[i] and in Z[w]

• Main Authors: Po-Ju Shih, 施柏如
• Other Authors: Liang-Chung Hsia
• Format: Others
• Language: en_US
• Published: 2004
• Online Access: http://ndltd.ncl.edu.tw/handle/66621790256839342117

碩士 === 國立中央大學 === 數學研究所 === 92 === 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.