Legendre's Theorem in Z[i] and in Z[w]
碩士 === 國立中央大學 === 數學研究所 === 92 === 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, squar...
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ndltd-TW-092NCU054790032015-10-13T13:04:43Z http://ndltd.ncl.edu.tw/handle/66621790256839342117 Legendre's Theorem in Z[i] and in Z[w] Legendre的定理在Z[i]和Z[w]的情形 Po-Ju Shih 施柏如 碩士 國立中央大學 數學研究所 92 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. Liang-Chung Hsia 夏良忠 2004 學位論文 ; thesis 28 en_US |
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碩士 === 國立中央大學 === 數學研究所 === 92 === 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.
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author2 |
Liang-Chung Hsia |
author_facet |
Liang-Chung Hsia Po-Ju Shih 施柏如 |
author |
Po-Ju Shih 施柏如 |
spellingShingle |
Po-Ju Shih 施柏如 Legendre's Theorem in Z[i] and in Z[w] |
author_sort |
Po-Ju Shih |
title |
Legendre's Theorem in Z[i] and in Z[w] |
title_short |
Legendre's Theorem in Z[i] and in Z[w] |
title_full |
Legendre's Theorem in Z[i] and in Z[w] |
title_fullStr |
Legendre's Theorem in Z[i] and in Z[w] |
title_full_unstemmed |
Legendre's Theorem in Z[i] and in Z[w] |
title_sort |
legendre's theorem in z[i] and in z[w] |
publishDate |
2004 |
url |
http://ndltd.ncl.edu.tw/handle/66621790256839342117 |
work_keys_str_mv |
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1717730961176657920 |