| 姓名 |
施柏如(Po-Ju Shih)
查詢紙本館藏 |
畢業系所 |
數學系 |
| 論文名稱 |
Legendre的定理在Z[i]和Z[w]的情形
(Legendre's Theorem in Z[i] and in Z[w])
|
| 相關論文 | |
| 檔案 |
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|
| 摘要(英) |
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. |
| 關鍵字(中) |
|
關鍵字(英) |
★ Legendre's Theorem |
| 論文目次 |
Introduction ...... 3
Preliminary:Fundamental Properties of Z[i] and Z[w]......5
Legendre's Theorem in Z[i]......14
Legendre's Theorem in Z[i]......18
Conclusion......24 |
| 參考文獻 |
Hungerford, T. W., Algebra, Springer-Verlag, New York, 1974.
Ireland, K. and Rosen, M., A Classical Introduction to Modern Number Thoery, Springer-Verlag, New York, 1982.
Snart, P. A., An equation in Gaussian integers, The American Mathematical Monthly, Vol. 59, No. 7, 448-254, 1952. |
| 指導教授 |
夏良忠(Liang-Chung Hsia)
|
審核日期 |
2004-1-16 |
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