課程名稱 |
代數二 Algebra (Ⅱ) |
開課學期 |
102-2 |
授課對象 |
理學院 數學系 |
授課教師 |
李白飛 |
課號 |
MATH7106 |
課程識別碼 |
221 U3840 |
班次 |
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學分 |
3 |
全/半年 |
半年 |
必/選修 |
選修 |
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 |
天數305天數305 |
備註 |
研究所數學組基礎課。先修知識:代數導論。 總人數上限:30人 外系人數限制:5人 |
Ceiba 課程網頁 |
http://ceiba.ntu.edu.tw/1022MATH7106_ |
課程簡介影片 |
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核心能力關聯 |
核心能力與課程規劃關聯圖 |
課程大綱
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課程概述 |
Rings and Modules: Exact Sequences. Projective and Injective Modules. Hom and Duality. Tensor Products. Algebras.
Fields: Field Extensions. The Fundamental Theorem. Splitting Fields, Algebraic Closure and Normality. The Galois Groups of a Polynomial. Finite Fields. Separability. Cyclic Extensions. Radical Extensions. Transcendental Bases.
Linear Algebra: Determinants. Decomposition of a Single Linear Transformation and Similarity.
Commutative Rings: Prime and Primary Ideals. Primary Decoposition. Noetherian Rings and Modules. Ring Extensions. Dedekind Domains. The Hilbert Nullstellensatz.
Non-Commutative Rings: Simple and Primitive Rings. The Jacobson Radical. Semisimple Rings. Algebras. Division Algebras. |
課程目標 |
A deeper survey of basic notions in algebra, possibly required in other graduate courses. |
課程要求 |
建議先修 ALGEBRA (I) MATH 7105.
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預期每週課後學習時數 |
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Office Hours |
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指定閱讀 |
Thomas W. Hungerford, "Algebra". |
參考書目 |
Nathan Jacobson, "Basic Algebra II".
Oscar Zariski and Pierre Samuel, "Commutative Algebra I". |
評量方式 (僅供參考) |
No. |
項目 |
百分比 |
說明 |
1. |
Midterm Examination |
50% |
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2. |
Final Examination |
50% |
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週次 |
日期 |
單元主題 |
第1週 |
2/17,2/20 |
Tensor Products. Algebras. |
第2週 |
2/24,2/27 |
Tensor Products of Algebras. Field extensions. Ruler and Compass Constructions. |
第3週 |
3/03,3/06 |
The Fundamental Theorem. Symmetric Rational Functions. |
第4週 |
3/10,3/13 |
Splitting Fields, Algebraic Closure and Normaility. |
第5週 |
3/17,3/20 |
Splitting Fields, Algebraic Closure and Normality (Continued). The Fundemental Theorem of Algebra. The Galois Group of a Polynomial. |
第6週 |
3/24,3/27 |
The Galois Group of a Polynomial (Continued). Finite Fields. Separability. |
第7週 |
3/31 |
Separability (Continued). |
第8週 |
4/07,4/10 |
Cyclic Extensions. Radical Extensions. |
第9週 |
4/14,4/17 |
Radical Extensions (Continued). The General Equation of Degree n. Transcendental Bases. Determinants. |
第10週 |
4/21,4/24 |
Midterm Examination. Determinants (Continued). Decomposition of a Single Linear Transformation. |
第11週 |
4/28,5/01 |
Rational and Jordan Canonical Forms. Prime, Maximal and Primary Ideals. |
第12週 |
5/05,5/08 |
Jacobson Radical. Local Rings. Noetherian Rings. |
第13週 |
5/12,5/15 |
Primary Decomposition in Noetherian Rings. |
第14週 |
5/19,5/22 |
Algebraic Variaties. Hilbert Nullstellensatz. Integral Dependence. |
第15週 |
5/26,5/29 |
Integers in Quadratic Fields. Simple and Primitive Rings. |
第16週 |
6/05 |
Nil and Nilpotent Ideals. Semisimple Rings. |
第17週 |
6/09,6/12 |
Semisimple Modules. |
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