博碩士論文 90221001 詳細資訊




以作者查詢圖書館館藏 以作者查詢臺灣博碩士 以作者查詢全國書目 勘誤回報 、線上人數:11 、訪客IP:34.204.191.31
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
★ 數論在密碼學上的應用★ a^n-b^n的原質因子,其中a,b為高斯整數
★ Group Representations on GL(2,F_q)★ Diophantine approximation and the Markoff chain
★ The average of the number of r-periodic points over a quadratic number field.★ 週期為r之週期點個數的平均值
★ 橢圓曲線上扭點的平均數★ 正特徵值函數體上的逼近指數之研究
★ On some problem in Arithmetic Dynamical System and Diophantine Approximation in Positive Characteristic★ ZCm 的理想環生成元個數之上限
檔案 [Endnote RIS 格式]    [Bibtex 格式]    [相關文章]   [文章引用]   [完整記錄]   [館藏目錄]   [檢視]  [下載]
  1. 本電子論文使用權限為同意立即開放。
  2. 已達開放權限電子全文僅授權使用者為學術研究之目的,進行個人非營利性質之檢索、閱讀、列印。
  3. 請遵守中華民國著作權法之相關規定,切勿任意重製、散佈、改作、轉貼、播送,以免觸法。

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
推文 facebook   plurk   twitter   funp   google   live   udn   HD   myshare   reddit   netvibes   friend   youpush   delicious   baidu   
網路書籤 Google bookmarks   del.icio.us   hemidemi   myshare   

若有論文相關問題,請聯絡國立中央大學圖書館推廣服務組 TEL:(03)422-7151轉57407,或E-mail聯絡  - 隱私權政策聲明