課程大綱

課程資訊

課程名稱

代數導論一
Introduction to Algebra (Ⅰ)

開課學期

101-1

授課對象

理學院數學系

授課教師

王姿月

課號

MATH2105

課程識別碼

201 24210

班次

學分

全/半年

半年

必/選修

必修

上課時間

星期一3,4(10:20~12:10)星期四7,8(14:20~16:20)

上課地點

新204新204

備註

教學改善計畫課程有教學助理實施小班輔導。時段:四8
總人數上限：100人
外系人數限制：10人

Ceiba 課程網頁

http://ceiba.ntu.edu.tw/1011abstactalgebraW

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課程概述

＊Contents：
1. Integers and Permutations: divisors and prime factorization, integers modulo n, permutations
2. Groups: groups, subgroups, cyclic groups, homomorphisms, cosets, Lagrange theorem, groups of motions and symmetries, isomorphism theorem
3. Rings: examples and basic properties, integral domains and fields, ideal and factor rings, homomorphisms, polynomial rings, symmetric polynomials, unique factorization domains, principal ideal domain

4. Fields: algebraic extensions, splitting fields, finite fields, geometric construction(ruler and compass), fundamental theorem of algebra

課程目標

Understand the structure of groups, rings, and fields.

課程要求

預期每週課後學習時數

Office Hours

每週五 15:00~16:00
每週一 15:00~16:00

指定閱讀

教科書: W. K. Nicholson, Introduction to Abstract Algebra, 3rd edition, John Wiley & Sons, 2007

參考書目

T. W. Judson, Abstract Algebra Theory and Applications
(can be obtained via http://abstract.pugetsound.edu/download.html

評量方式
(僅供參考)

No.	項目	百分比	說明
1.	作業	20%	Homework will be collected every Thursday (tutorial class). No Late Homework!
2.	平時考	20%	(Thursday tutorial class) 20 minutes for each test, total of 7-10 tests for this semester
3.	期中考	30%	the Week of November 4
4.	期末考	30%	the Week of January 7

課程進度

週次	日期	單元主題
第1週	9/10,9/13	* Definition and example of groups, rings, and fields * Division Algorithm, Euclidean Algorithm equivalence relation Integers modulo n
第2週	9/17,9/20	Chinese remainder theorem, Fermat's theorem, an application to crytography, permutations, binary operations
第3週	9/24,9/27	definition of groups, basic properties of groups, classification of groups with 2,3,or 4 elements, subgroups
第4週	10/01,10/04	examples of subgroups, cyclic groups, order of an element, fundamental theorem of finite cyclic groups, examples of group homomorphism and group isomorphism
第5週	10/08,10/11	basic properties of homomorphism and isomorphism, automorphism, inner automorphism, Cayley's theorem, cosets, Langrange's theorem
第6週	10/15,10/18	Application of Lagrange's theorem, Dihedral groups, classification of groups of order 6, Euler's theorem, normal subgroups
第7週	10/22,10/25	classification of order 8 abelian groups, definition of factor groups (quotient groups) and basic properties, commutator subgroups (derived groups), alternation groups (A_n)
第8週	10/29,11/01	kernel, image, isomorphism theorem, homomorphic images of a group, alternating groups (A_n)
第9週	11/05,11/08	simple groups, examples of simple groups, more on alternating groups, review for midterm exam
第10週	11/12,11/15	definition of rings, examples, basic properties, characteristic of a ring, subrings, units, idempotents, nilpotents, division rings, fields
第11週	11/19,11/22	integral domains, fields, field of quotients, ideals, factor rings
第12週	11/26,11/29	prime ideals, maximal ideals, ring homomorphisms, isomorphism theorem
第13週	12/03,12/06	applications of the isomorphism theorem, decomposition of rings, Chinese remainder theorem, polynomial rings, division algorithm of division ring, evaluation theorem, remainder theorem
第14週	12/10,12/13	factorization of polynomial rings over fields, Gauss lemma, Eisenstein criterion
第15週	12/17,12/20	factor rings of polynomials over a field, Kronecker's theorem, algebraic extension of a field, minimal polynomial of an algebraic element over a field
第16週	12/24,12/27	splitting fields, finite fields
第17週	12/31,1/03	geometric construction(ruler and compass), fundamental theorem of algebra