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    Please use this identifier to cite or link to this item: http://ir.lib.ncu.edu.tw/handle/987654321/63707


    Title: Legendre的定理在Z[i]和Z[w]的情形;Legendre's Theorem in Z[i] and in Z[w]
    Authors: 施柏如;Shih,Po-Ju
    Contributors: 數學研究所
    Keywords: Legendre's Theorem
    Date: 2004-01-16
    Issue Date: 2014-05-08 15:26:30 (UTC+8)
    Publisher: 國立中央大學
    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.
    Appears in Collections:[數學研究所] 博碩士論文

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