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研究生: 藍茂愷
Mao-Kai Lan
論文名稱: 數論中的局部-全域原則
The Local-Global Principle in Number Theory
指導教授: 夏良忠
Hsia, Liang-Chung
學位類別: 碩士
Master
系所名稱: 數學系
Department of Mathematics
論文出版年: 2014
畢業學年度: 102
語文別: 英文
論文頁數: 31
中文關鍵詞: 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
英文關鍵詞: Local-Global Principle, Hasse Principle, Quadratic Reciprocity Law, Hilbert Symbol, Hasse-Minkowski Theorem, Sum of Squares
論文種類: 學術論文
相關次數: 點閱:143下載:23
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  • 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.

    Contents 1. Introduction 01 2. Priliminaries 03 3. Basic Results on Quadratic Spaces and Quadratic Forms 06 4. Hilbert Symbol 10 5. Hasse-Minkowski Theorem for Q and Its Applications 16 6. Hasse Principle for Powers 22 7. Hasse-Minkowski Theorem for F_p (t) with p≠2 23 8. Some Counterexamples to the Hasse Principle 29 9. References 31

    [1] Atiyah, M.F. and Macdonald, I.G. Introduction to Commutative Algebra, Addison-Wesley, 1969.
    [2] Cohen, H. Number Theory Volume I: Tools and Diophantine Equations, GTM 239, Springer-Verlag, 2007.
    [3] Fulton, W. Algebraic Curves; An Introduction to Algebraic Geometry, W.A. Benjamin, 1969.
    [4] Janusz, G.J. Algebraic Number Fields, Acadmic Press, 1973.
    [5] Lang, S. Algebra, GTM 211, Springer-Verlag, 2002.
    [6] Neukirch, J. Algebraic Number Theory, Grundl. Math. Wissens. 322, Springer-Verlag, 1999.
    [7] Rosen, M. Number Theory in Function Fields, GTM 210, Springer-Verlag, 2002.
    [8] Serre, J. P. A Course in Arithmetic, GTM 7, Springer-Verlag, 1973.
    [9] Silverman, J. H. The Arithmetic of Elliptic Curves, 2nd Edition, GTM 106, Springer-Verlag, 2008.

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