近世代數初探
課程編號: SAYT1014
上課日期: 15/7, 16/7, 19/7, 20/7, 22/7, 23/7, 26/7, 27/7, 28/7, 2021
上課時間: 10:30am – 5:15pm
考試日期: 30/7/2021 (10:30am – 1:30pm)
後備課堂日期: 21/7, 29/7, 31/7, 2021 (10:30am-5:15pm)
上課地點: [中文大學, 詳情有待公佈]*
任教導師: 李俊捷博士(香港中文大學)
大學認可: 中文大學1學分
教學語言：粵語為主，附以英文教材
對象: 準備升上中四或中五的同學參加，學員須具備良好的抽象數學推理能力。
內容簡介: 本科將從經典代數開始，討論如何使用根式解三次方程及四次方程、使用二分法求解方程、圓規直尺作圖問題、利用圓規直尺構造正則十七邊形、根與係數之間的關係、對稱多項式、求 k 次和公式等。接著將引入抽象代數結構，包括群、域、向量空間，並為同學介紹具體的數學對象，包括複數、二次域、四元數、多項式、模為 n 的加法和乘法群、置換群、橢圓曲線、有限域等，來說明這些代數結構。
學費: 港幣 3,990元正 (已包含港幣40元申請費用)
備註:
*此課程暫定於中文大學內授課，或有可能轉為線上授課，視乎實際情況而定。
**如課程最終轉為線上授課，學費將會調整為港幣 3,040元正 (已包含港幣40元申請費用)。
Towards Modern Algebra
Course code: SAYT1014
Date: 15/7, 16/7, 19/7, 20/7, 22/7, 23/7, 26/7, 27/7, 28/7, 2021
Time: 10:30am – 5:15pm
Examination Date: 30/7/2021 (10:30am – 1:30pm)
Reserved Date: 21/7, 29/7, 31/7, 2021 (10:30am-5:15pm)
Venue: [CUHK. Detail to be announced]*
Lecturer: Dr. LI Chun Che (CUHK)
University Recognition: 1 credit of CUHK
Medium of Instruction: mainly in Cantonese with English course materials
Expected applicants: Students who have high competence in abstract mathematical reasoning, and are promoting to Secondary 4 or Secondary 5
Introduction: The central themes of classical algebra includes the study of polynomials, finding roots of polynomials, solving system of equations, Compass-and-straightedge construction. In response to these problems, modern abstract algebra was introduced during 19th century.
In this course, we start with classical algebra topics which include using radicals to solve cubic and biquadratic equations, bisection method, compass-and straightedge constructions, construction of regular-17-gon, relationship between roots and coefficients, symmetric polynomials, closed formula of sum of powers, among other things.
We will then introduce abstract algebraic structures including groups, fields, vector spaces with emphasis on concrete examples. We will introduce concrete math objects, including complex numbers, quadratic fields, quaternions, polynomials, additive and multiplicative group mod n, permutation groups, elliptic curves, finite fields, among other things, to illustrate the concepts of the algebraic structures.
Tuition fee: HK$3,990 (with $40 application fee included)**
Remark:
*This course offers face-to-face lessons on CUHK campus. It may switch to online teaching, subject to the actual situation.
**The tuition fee will be adjusted to HK$3,040 (with $40 application fee included) if the course finally switches to online teaching.
網上報名(Online application)
入學試免試條款 Conditions for Admission Screening Test Exemption
凡符合以下其中一項條款的申請人，可免除入學試而獲本科直接取錄。（參考附註）
Applicants who satisfy either one of the following conditions may be exempted from Admission Screening Test and will be directly admitted into this course. (refer to note below)
- 曾修讀並及格完成以下任何一個科目取錄: 「複數的幾何面貌」、「複數與解析幾何」、「近世代數初探」。
Passed in any of the following courses before: Geometric Perspectives of Complex Numbers, Complex Number and Analytical Geometry, and Towards Modern Algebra.
- 曾獲得以下任何一個科目取錄:「數論與密碼學」、「微分幾何初探」、「數學分析入門」、「非歐幾何賞析」。
Being admitted in one of the following courses before: Number Theory and Cryptography, Towards Differential Geometry, Mathematical Analysis, and Understanding Non-Euclidean Geometry.
（參考附註 note）
曾修讀「數學啟導修習I或II」的學生，仍須參加入學試，但可獲優先考慮。
Those who had taken "Enrichment Mentoring Mathematics I or II" still need to sit the Test, but will be considered with priority.
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