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研究生： 陳慧錚
Huei Jeng Chen
論文名稱： 在Q_p 上維度為2 和 3 的體擴張
Extensions of Q_p of degree 2 and 3
指導教授： 李華介
Li, Hua-Chieh
學位類別： 碩士
Master
系所名稱： 數學系
Department of Mathematics
論文出版年： 2005
畢業學年度： 93
語文別： 中文
論文頁數： 45
中文關鍵詞： 維度 、擴張
論文種類： 學術論文
相關次數： 點閱：190  下載：10 
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    • 大家都知道對於Q_p來說, 給定一個維度n必只存在有限多個在Q_p上擴張體維度是n的. 對於每個正整數n, 我們觀察係數在Q_p這個體上的多項式而且最高次是n次它的根的行為. 我們在這篇論文中主要是要研究多項式的係數和體擴張之間的關係.

    It is well-known that for a p-adic
    field there exists only finite many extensions of a given degree.
    For a polynomial f with coefficients satisfying some conditions
    in Qp of given degree and let alpha be any root of
    f, we want to discuss the connection between coefficients of f
    and types of extensions Qp(alpha). In this paper we
    present a method for discussing the relation.

    Chapter 1. Introduction 1 Chapter 2. Notations and Preliminary results .............3 2.1. Definitions and properties ..........................3 2.2. Unramified extensions of Qp .........................6 2.3. Totally ramified extensions of Qp ...................8 2.4. Krasner’s Lemma ...................................10 2.5. Discriminant .......................................14 Chapter 3. The number of totally ramified extensions of Qp of degree n .............................................19 3.1. Eisenstien polynomials and totally ramified extensions ..............................................19 3.2. Calculate the number of totally ramified extensions of Qp ......................................................23 Chapter 4. Finite extensions of Qp of degree 2 ..........27 4.1. Irreducible polynomials of degree 2 ................27 4.2. The case p =\= 2 ...................................29 4.3. The case p = 2 .....................................32 Chapter 5. Finite extension of Qp of degree 3 ...........35 5.1. Irreducible polynomials of degree 3 ................35 5.2. The case p =\= 3 ...................................35 5.3. The case p = 3 .....................................37 Bibliography ............................................41

    (1)I.B. Fesenko and S.V. Vostokov,
    "Local Fields and Their Extensions", 2nd ed., With a
    foreword by I. R. Shafarevich. Translations of Mathematical
    Monographs, 121. American Mathematical Society, Providence, RI,2002

    (2)Neal Koblitz, "p-adic Numbers, p-adic Analysis,
    and Zeta-Functions", 2nd ed., Springer- Verlag, Berlin
    and New York, 1984

    (3)Thomas W. Hungerford,"Algebra",
    Springer-Verlag, New York, 1974

    (4)Serge Lang, "Algebraic Number Theory",
    Spriger-Verlag, Berlin and New York, 1986

    (5)Sebastian Pauli and Xavier-Francois
    Roblot, "On the computation of all extensions of p-adic
    field of a given degree, Mathematics Of Computation, Volume 70,
    Number236, page 1641-1659, 2001

    (6)Gerald J. Janusz, "Algebraic Number
    Fields, 2nd ed.

    (7)Ore, Bemerkungen zur Theorie der Differente,
    Math. Zeischr. 25 (1926), pp.1-8.

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