課程大綱

課程資訊

課程名稱

代數導論二
Introduction to Algebra(Ⅱ)

開課學期

106-2

授課對象

理學院數學系

授課教師

林惠雯

課號

MATH2114

課程識別碼

201 49620

班次

學分

4.0

全/半年

半年

必/選修

必修

上課時間

星期一3,4(10:20~12:10)星期四8,9(15:30~17:20)

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。
限本系所學生(含輔系、雙修生)
總人數上限：80人

Ceiba 課程網頁

http://ceiba.ntu.edu.tw/1062MATH2114_Basic2

課程簡介影片

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課程大綱

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課程概述

I. Ring theory:
Polynomial rings, Ideals, Localization, Hilbert's Nullstellensatz, Grobner basis, ED + PID + UFD, Algebraic integers, Ideal classes, Class group, Quadratic number fields

II. Module theory:
Free modules, Diagonalizing integer matrices, Finitely generated modules over a PID, Structure theorems

III. Galois theory:
Finite extensions, Algebraic extensions, Ruler and compass constructions, finite fields, Primitive element theorem, Splitting fields, Galois extensions, Fundamental theorem of Galois theory, Cubic and Quartic equations, Cyclotomic extensions, Kummer extensions, Quintic equations

課程目標

Algebra structures appear as common abstract structures of underlying symmetries in various areas of mathematics, and conversely the language and concepts of algebra have integrated naturally into all modern mathematics, both pure and applied, as useful tools to describe various phenomena in a systematical way. The goal of the course is to equip students the knowledges of the most basic objects (groups, rings, modules and fields) in abstract algebra. After the course, we hope that students shall feel comfortable in using language from algebra in all areas of sciences.

課程要求

Course on "Introduction to Algebra (I)"

預期每週課後學習時數

Office Hours

備註：星期二中午 12:20 ~ 1:10

指定閱讀

M. Artin, Algebra, 2rd edition, 2010.

參考書目

N. Jacobson, Basic Algebra I , 2nd edition
Dummit-Foote, Abstract Algebra
Serge Lang, Undergraduate Algebra, 3rd edition

評量方式
(僅供參考)

No.	項目	百分比	說明
1.	期中考	30%
2.	期末考	30%
3.	小考	20%	共兩次,每次 10 %
4.	作業	20%	由助教評分

課程進度

週次	日期	單元主題
第1週	2/26,3/01	Ideals and Isomorphism theorems / Chinese remainder theorem ; Chapter 11 : 3.9, 3.10, 3.11, 4.3, 5.5 / Chapter 11 : 6.1, 6.7, 8.2
第2週	3/05,3/08	Localization / Hilbert Nullstellensatz ; Chapter 11 : 7.1, 7.2, 7.3, 7.4 / Chapter 11 : 8.1, 8.3, 8.4
第3週	3/12,3/15	Grobner basis (I) / Grobner basis (II) ; 3/9上傳資料的 19, 20, 21, 23 / Chapter 11 : 9.1, 9.7, 9.12
第4週	3/19,3/22	ED +PID + UFD (I) / ED + PID + UFD (II) ; Chapter 12 : 2.1, 2.2, 2.4, 2.9 / Chapter 12 : 2.10, M.7
第5週	3/26,3/29	Gauss's lemma and Irreducibility / Gauss primes ; 小考 ; Chapter 12 : 3.2, 3.6, 4.4, 4.5, 4.18 / Chapter 12 : 5.1, 5.3, 5.6
第6週	4/02,4/05	溫書假 / 掃墓節放假
第7週	4/09,4/12	Algebraic integers (I) / Algebraic integers (II) ; Chapter 13 : 1.3, 3.2, 3.3, 4.3, 5.2 / Chapter 13 : 5.1, 6.1, 6.3
第8週	4/16,4/19	Ideal classes and class group (I) / Ideal classes and class group (II) ; Chapter 13 : 7.1, 7.3, 7.4, 10.1 / Chapter 13 : 8.3
第9週	4/23,4/26	Free modules / 期中考
第10週	4/30,5/03	Matrices over a PID / Structure of finitely generated modules over a PID ; Chapter 14 : 4.1, 4.4, 4.6, 4.7 / Chapter 14 : 7.3, 7.5, 7.7
第11週	5/07,5/10	Applications / Simple extensions ; Chapter 14 : 8.1, 8.2, 8.3 / Chapter 15 : 1.1, 1.2, 1.3, 2.1
第12週	5/14,5/17	Finite extensions and Algebraic extensions / Ruler and compass constructions ; Chapter 15 : 3.1, 3.7, 3.8, 4.1, 4.2 / Chapter 15 : 5.2, 5.3, 5.4
第13週	5/21,5/24	Algebraic closure and Primitive element theorem / Extensions of a finite field ; 小考 ; Chapter 15 : 6.1, 6.2, 6.3, 8.2, 10.1 / Chapter 15 : 7.5, 7.8, 7.12
第14週	5/28,5/31	Splitting fields and Fixed fields / Galois extensions and Galois groups ; Chapter 16 : 1.3, 3.2, 4.1, 5.2 / Chapter 16 : 6.1, 6.2, 6.3
第15週	6/04,6/07	Fundamental theorem of Galois theory / Cyclotomic extensions ; Chapter 16 : 7.2, 7.4, 7.8, 7.10, 7.11 / Chapter 16 : 10.1, 10.3, 10.5
第16週	6/11,6/14	Kummer extensions / Solution by radicals (I)
第17週	6/18,6/21	端午節放假 / Solution by radicals (II)
第18週	6/25	期末考