基本素養 Basic Literacy

核心能力 Competence

掌握連續與逼進的能力
Ability to handle continuity phenomena and approximations.
處理數字及符號運算的能力
Ability to operate with symbols and digits.
透視形像與空間的能力
Ability to visualize space, shape, and images.
利用計算機處理數學的能力
Ability of using computer to do mathematics.
應用隨機理論的能力
Ability to apply stochastic mathematics.
處理大量數據的能力
Ability to handle large amount of data.
處理離散型數學的能力
Ability to do mathematics relating to discrete models.
獨立研究的能力
Ability to do independent studies of mathematics.
使用計算機的能力
Abilities to use computer as a tool.

課程概述 Course Description

代數學(一)及(二)延續了線性代數課程,但是其題材更為抽象,也有更多證明的訓練。在此一課程中,會介紹群、環、體等結構,發展研究其結構的方法,並且將其應用在不同的領域中。此一課程會持續地帶領學生深入了解數學,也增強學生在以數學符號及口頭來溝通數學,讓學生能優游自在地閱讀、了解數學,進一步培養對抽象數學的鑑賞能力。
Algebra (1) and (2) is a natural continuation of Linear Algebra, but the material is much more proof-driven and abstract. During the year, the structures groups, rings, and fields will be introduced, and the method of studying them will be developed, and applications to other areas will be seen. The general objective of this course is to continue providing students with a deeper understanding and working knowledge of mathematics, strengthening the analytic skills, increasing the ability to communicate mathematics symbolically and orally, making students comfortable with reading and understanding mathematics on their own, and continuing to develop the appreciation for abstract mathematics.

課程學習目標 Course Objectives

  • 介紹代數結構及其研究方法。
  • 學習如何證明。
  • 了解群 (groups)、環 (rings) 結構的基本性質及例子。
  • 課程進度 Course Outline

    週次 Week進度說明 Progress Description
    1Set Theory, Mappings, Integers
    2Mathematical Inductions, Complex Numbers
    3Definition and Examples of Groups
    4Subgroups and Lagrange's Theorem
    5Homomorphisms and Normal Subgroups
    6Homomorphisms and Normal Subgroups
    7Factor groups
    8The Isomorphism Theorems
    9期中考
    10Cauchy's Theorem
    11Direct Products
    12Finite Abelian Groups
    13Conjugacy and Sylow's Theorem
    14Conjugacy and Sylow's Theorem
    15Permutation Groups
    16Permutation Groups
    17Permutation Groups
    18期末考
     以上每週進度教師可依上課情況做適度調整。The schedule may be subject to change.

    有關課程其他調查 Other Surveys of Courses

    1.本課程是否規劃業界教師參與教學或演講? 否
    Is there any industry specialist invited in this course? How many times? No
    2.本課程是否規劃含校外實習(並非參訪)? 否
    Are there any internships involved in the course? How many hours? No
    3.本課程是否可歸認為學術倫理課程?
    Is this course recognized as an academic ethics course? In the course how many hours are regarding academic ethics topics?
    4.本課程是否屬進入社區實踐課程?
    Is this course recognized as a Community engagement and Service learning course? Which community will be engaged?

    教師上傳大綱內容
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    Syllabus-1.pdf