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研究生： 南婷婷
Ting-Ting, Nan
論文名稱： 函數體上的算術
On the Arithmetic of Function Fields
指導教授： 許志農
Hsu, Chih-Nung
學位類別： 碩士
Master
系所名稱： 數學系
Department of Mathematics
論文出版年： 2009
畢業學年度： 97
語文別： 英文
論文頁數： 46
中文關鍵詞： 特徵函數 、多項式環
英文關鍵詞： character, polynomial ring
論文種類： 學術論文
相關次數： 點閱：212  下載：8 
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    • 這篇論文包含三個章節, 都是討探在係數為有限體的多項環上的算術問題. 此三個問題分別為:
    On the Polynomial Gauss Sums(mod P^n),n>1,
    On the Polynomial Schur's Matrix,
    和
    On the Number of Solutions of Linear Equation in Finite Carlitz Modules.

    1 On Polynomial Gauss Sums (mod P^n), n>1...............1 1.1 Introduction............... 1 1.2 Auxiliary Lemmas ............... 3 1.3 The Arguments of Polynomial Gauss Sums ............... 7 1.4 Application to polynomial Chowla-Mordell Theorem ............... 16 2 On the Polynomial Schur’s Matrix ............... 19 2.1 Introduction ............... 19 2.2 The Multiplicity of the Eigenvalues ............... 21 2.3 The Eigenvectors of the Polynomial Schur’s Matrix ............... 23 3 On the Number of Solutions of Linear Equation in Finite Carlitz Modules ............... 31 3.1 Introduction ............... 31 3.2 Auxiliary Lemmas ............... 33 3.3 The main theorem ............... 37 3.4 Application to normal bases ............... 40

    Chapter 1
    [1] S. Chowla, On Gaussian Sums, Proc. Nat. Acad. Sci. USA 48 (1962), 1127–1128.
    [2] R. J. Evans, Generalizations of a Theorem of Chowla on Gaussian Sums, Houston J. Math. 3(1977), 343-349.
    [3] R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press(1997).
    [4] C.-N. Hsu, On Polynomial Reciprocity Law, J. Number Theory 101(2003) 13–31.
    [5] L. J. Mordell, On a Cyclotomic Resolvent, Arch. Math. 13 (1962), 486–487.
    [6] R. Odoni, On Gauss Sums (mod pn), n  2, Bull. London Math. Soc. 5 (1973), 325–327.
    [7] J. Yang and W. Zheng, On a Theorem of Chowla, J. Number Theory 106(2004), 50-55.

    Chapter 2
    [1] L. Carlitz, The Singular Series for Sums of Squares of Polynomials, Duke Math. J. 14 (1959) 293–308.
    [2] L. Carlitz, Some Cyclotomic Matrices, Acta Arith. 5 (1947) 1105–1120.
    [3] C.-N. Hsu, On Polynomial Reciprocity Law, J. Number Theory 101 (2003) 13–31.
    [4] P. Morton, On the Eigenvectors of Schur’s Matrix, J. Number Theory 12 (1980) 122–127.

    Chapter 3
    [1] L. Carlitz, Sums of Primitive Roots in a Finite Field, Duke Math. J., 19 (1952), 459–469.
    [2] L. Carlitz, Sets of Primitive Roots, Compositio Math., 13 (1956), 65–70.
    [3] D. Goss, Basic Structures of Function Fields Arithmetic, Springer-Verlag (1996).
    [4] H. W. Lenstra and R. J. Schoof, Primitive Normal Bases for Finite Fields, Math. Comp. 48 (1987), 217–231.
    [5] R. Lidl and H. Niederreiter, Finite Fields, Cambridge University Press(1997).

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