數論中的局部-全域原則
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2014
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Abstract
在這篇論文中，我們探討算術當中最重要的課題之一：局部-全域原則。 作為第一個例子，我們將呈現著名的有理域 Q 上之哈瑟-閔可夫斯基定理及其幾個應用。 第二個例子則是乘冪上的哈瑟原則。 第三個例子我們欲討論當 p 不等於 2 時，函數域 F_p(t) 上之哈瑟-閔可夫斯基定理。 最後，我們將研究局部-全域原則的幾個反例。
In this thesis, we investigate one of the most important topics in arithmetic: the local-global principle. As a first example, we will present the well-known Hasse-Minkowski Theorem for Q and give some of its applications. The second example will be the Hasse principle for powers. The third one, we would like to discuss the Hasse-Minkowski Theorem for F_p(t) with p≠2. Finally, we will study some counterexamples to the local-global principle.
In this thesis, we investigate one of the most important topics in arithmetic: the local-global principle. As a first example, we will present the well-known Hasse-Minkowski Theorem for Q and give some of its applications. The second example will be the Hasse principle for powers. The third one, we would like to discuss the Hasse-Minkowski Theorem for F_p(t) with p≠2. Finally, we will study some counterexamples to the local-global principle.
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局部-全域原則, 哈瑟原則, 二次互反律, 希爾伯特符號, 哈瑟-閔可夫斯基定理, 平方和, Local-Global Principle, Hasse Principle, Quadratic Reciprocity Law, Hilbert Symbol, Hasse-Minkowski Theorem, Sum of Squares