| 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. |
