Abstract: | 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. |