代数学-1

Syllabuses on Algebra and Number Theory

Linear Algebra 

Abstract vector spaces; subspaces; dimension; matrices and linear transformations; matrix algebras and groups; determinants and traces; eigenvectors and eigenvalues,  characteristic and minimal polynomials; diagonalization and triangularization of operators;  invariant subspaces and canonical forms; inner products and orthogonal bases;  reduction of quadratic forms; hermitian and unitary operators, bilinear forms; dual spaces; adjoints. tensor products and tensor algebras; 

Integers and polynomials

Integers, Euclidean algorithm,  unique decomposition; congruence and the Chinese Remainder theorem; Quadratic reciprocity ; Indeterminate Equations. Polynomials, Euclidean algorithm, uniqueness decomposition, zeros;  The fundamental theorem of algebra; Polynomials of integer coefficients, the Gauss lemma and the Eisenstein criterion; Polynomials of several variables, homogenous and symmetric polynomials, the fundamental theorem of symmetric polynomials.

Group 

Groups and homomorphisms, Sylow theorem,  finitely generated abelian groups. Examples: permutation groups, cyclic groups, dihedral groups, matrix groups, simple groups, Jordan-Holder theorem, linear groups (GL(n, F) and its subgroups), p-groups, solvable and nilpotent groups, group extensions, semi-direct products, free groups, amalgamated products and group presentations.

Ring

Basic properties of rings, units, ideals, homomorphisms, quotient rings, prime and maximal ideals, fields of fractions, Euclidean domains, principal ideal domains and unique factorization domains, polynomial and power series rings, Chinese Remainder Theorem, local rings and localization, Nakayama’s lemma, chain conditions and Noetherian rings, Hilbert basis theorem, Artin rings, integral ring extensions, Nullstellensatz, Dedekind domains,algebraic sets, Spec(A).

Module  

Modules and algebra  Free and projective; tensor products; irreducible modules and Schur’s lemma; semisimple, simple and primitive rings; density and Wederburn theorems; the structure of finitely generated modules over principal ideal domains, with application to abelian groups and canonical forms; categories and functors; complexes, injective modues, cohomology; Tor and Ext. 

Field  

Field extensions, algebraic extensions, transcendence bases; cyclic and cyclotomic extensions; solvability of polynomial equations; finite fields; separable and inseparable extensions; Galois theory, norms and traces, cyclic extensions, Galois theory of number fields, transcendence degree, function fields.

 Group representation

 Irreducible representations, Schur’s lemma, characters, Schur orthogonality, character tables, semisimple group rings, induced representations, Frobenius reciprocity, tensor products, symmetric and exterior powers, complex, real, and rational representations.

 Lie Algebra

Basic concepts, semisimple Lie algebras, root systems, isomorphism and conjugacy theorems, representation theory. 

Combinatorics (TBA)

References:

1. Strang, Linear algebra, Academic Press.
2. I.M. Gelfand, Linear Algebra

1. 《整数与多项式》冯克勤 余红兵著 高等教育出版社
2. Jacobson, Nathan Basic algebra. I. Second edition. W. H. Freeman and Company, New York, 1985. xviii+499 pp.

1. Jacobson, Nathan Basic algebra. II. Second edition. W. H. Freeman and Company, New York, 1989. xviii+686 pp.

1. S. Lang, Algebra, Addison-Wesley
2. 冯克勤，李尚志，查建国，章璞，《近世代数引论》
3. 刘绍学，《近世代数基础》
4. J. P. Serre, Linear representations of finite groups
5. 10.J. P. Serre: Complex semisimple Lie algebra and their representations

1. 11.J. Humphreys: Introduction to Lie algebra and representation theory, GTM 009.

1. 12.W. Fulton, Representation theory, a First Course, GTM 129. 

1， 代数学简史、线性方程组、矩阵，把线性方程组与矩阵化为阶梯形，Gauss消去法、低阶行列式、集合与映射、二元关系、等价关系、商映射、偏序集。

2， 数学归纳法、置换、置换的循环结构、置换的符号、斜对称函数、数论的基本概念、算术基本定理。

3， 向量与纯量、线性组合、线性相关与线性无关、直线的线性无关性，线性无关的基本引理，基与维数、线性方程组的基与秩，矩阵的秩、利用矩阵的秩研究线性方程组的相容性，线性方程组的可解性准则、线性映射、线性变换、线性函数、矩阵的运算、逆矩阵、矩阵的等价类、线性方程组的基本解系。线性方程组的解空间。

3。有限集合上的置换，置换的奇偶性，交换群，置换分解为轮换的积。

4，方阵的行列式，作为有向体积的行列式、 行列式的基本性质、子式、余子式、行列式的展开。

5， 非退化行列式的判定、行列式值为零的判据，矩阵按行（列）展开，伴随矩阵、线性方程组的Cramer法则、加边子式法、作为多

重线性规范反对称函数的行列式。矩阵的秩定理，范德蒙德行列式。

5。矩阵的运算及其性质，逆矩阵，求拟矩阵的方法，基本变换，矩阵的基本变换和基本矩阵的乘积的关系。

6。基本代数系统：群、环、域，剩余类环，域的特征。

6， 二元运算、半群、幺半群、群、子群、循环群、群的同构、Cayley定理、群的同态与自同态、环、同余类、剩余类环、环的同态、整环、域、域的同构与自同构、域的特征、素域、复数域、复数的几何表示，本原根、复数的几何、交比。复数的三角和代数形式。

7，域上单变量多项式环，一元多项式环、多元多项式环、多项式分解成有余式的非平凡多项式之积的可能性与唯一性，唯一析因环、两个多项式的最大公约式，环中的最大公因与最小公倍、环中元素的互素、整除性的判定、欧几里德算法，Euclid环、既约多项式、本原多项式、Gauss引理、Eisentein判别法。

8。 整数域与多项式域的分裂，实数域与复数域上的不可约多项式。

9。多项式的根，形式导数及其性质，多项式根的标准分解式，多项式根的范围，笛卡尔定理，计算多项式实根数量的斯图姆方法。

8， 整环的分式域、有理函数域、最简分式、Bezout定理、多项式函数环、插值多项式，Lagrange多项式与Newton插值公式、多项式环的微分法、Vieta公式、对称与斜对称函数、Wilson定理。

9， 对称多项式环、多称多项式的基本定理、待定系数法、等幂和、Newton公式、多项式的判别式、结式、复数域的代数封闭性、代数基本定理、计算多项式实根数量的Strum定理、多项式根的近似算法、整系数多项式的有理根。

11。有理分数域，有理分实式分解成部分分式之和，实数与复数域上的情况。

12。多变量多项式环，对称多项式，他的基本对称多项式展开，维特定理。

13。循环群，他的子群，循环群的同构，群的分解的拉格朗日定理。

14。多项式的判别式，多项式的判别式与他们的根的关系。

10， 一般域上的线性空间、子空间、线性相关、线性无关、向量组的秩、基与维数、不同基之间的过渡矩阵、线性空间的同构、子空间的交与和、维数定理、直和、补空间、商空间、线性函数、对偶空间、线性无关的判别法。

11， 线性映射、线性映射的矩阵表示、像与核、线性算子、线性算子代数、极小多项式、矩阵的相似、线性算子的行列式与迹。

12， 不变子空间、特征值与特征向量、特征多项式、特征子空间、几何重数与代数重数、可对角化算子的判别法、不变子空间的存在性、共轭线性算子、商算子。

代数学-2

1， 范畴、函子、Hamilton-Cayley定理、Jordan标准型、根子空间、循环子空间、循环矩阵、矩阵的有理标准型。

2， 多项式矩阵、多项式矩阵的初等变换、多项式矩阵的相抵、Smith标准型、行列式因子、不变因子、初等因子组、特征方阵与Jordan标准型的关系、实方阵的实相似。

3， 多重线性映射、双线性型、矩阵的相合变换、双线性型的秩、左根基、对称双线性型与斜对称双线性型、二次型、二次型的规范型、化二次型为规范型的方法、实二次型、惯性定理、正定二次型与正定矩阵、Jacobi方法、Sylvester定理、斜对称二次型的规范型、Pfaff型。

4， Euclid空间、内积、标准正交基、Gram-Schmidt正交化过程、Euclid 空间的同构、正交矩阵、正交群、辛空间、辛群、辛算子、酉空间、Hermite型、酉矩阵、酉群、赋范线性空间、按模收敛、绝对收敛。

5， 内积空间上的线性算子、化二次型为主轴形式、把两个二次型同时化为规范型、保距算子的规范形式、极分解、奇异值分解、Schur定理、Witt扩张定理、复结构、复化线性空间、实化线性空间、实化线性算子、复化算子、最小二乘法、球面多项式、加权正交。

6， 线性算子的范数、线性群的单参数子群、谱半径、仿射空间、仿射映射、仿射空间的同构、仿射子空间、仿射坐标系、仿射同构、Euclid度量、Gram行列式、有向体积。

7， 仿射群、Euclid空间的运动群、保距变换群、凸集、Minkowski空间、伪欧氏空间、Lorenz群、仿射空间上的二次函数、化二次函数为规范型、Euclid空间上的二次函数。

8， 二次曲面、二次曲面的中心、仿射空间中二次曲面的规范型、二次曲面

的分类、Euclid空间中的二次曲面、射影平面、高维射影空间、齐次坐标、仿射几何与射影几何的关系、代数簇、射影群、交比与重比、射影空间中二次曲面的分类、直线与射影二次曲面的相交。

9， 张量的概念、张量的坐标、张量积、张量的卷积、对称与斜对称张量、张量空间、外代数。

1。群，子群，群的同构，群的阶，循环群，对称群的生成集，交错群与线性群，凯莱定理。

10， 正规子群、左陪集与右陪集、代表元、Lagrange定理、循环群的结构、群在集合上的作用，群作用、轨道、稳定子群与轨道，稳定子群、群的共轭类。正规化子、可迁群、齐次空间。

11， 典型群、满同态、四元数代数、置换群、对称。

12， 商群、群的同态，同态基本定理、群的同构定理、

3。环的理想，商环，环的同太定理，多项式的分裂域，有限域。

换位群，换位子群、群的直积与半直积、生成元、自由群、可解群、有限R-群与对角矩阵群的可解性，有限阶交错群。单群。

1， Zassenhaus引理、Jordan-Holder定理、带算子的群、自同态环、自同构类群、Sylow定理、特征子群、Abel群、自由阿贝尔群，有限生成的Abel群、Frobenius-Stickelberger定理、有限Abel群的基本定理。有限生成阿贝尔群的结构定理。

7。域上的代数，实数域上的弗罗比尼斯定理，李代数。

8。有限群的表示论初步，特征标，可约表示与完全可约表示，有限群的完全可约表示的马舒克定理，群的正则表示理论，特征标的正则性，不可约表示的维数。

代数学-3

2， 良序集、Zorn引理、选择公理、态射、自然变换、环的理想、商环、同态基本定理、环的同构定理、理想的运算、局部化、素理想。

3， Gauss整数、主理想环、

1。主理想环与Euclid环，主理想环与Euclid环上有限模的结构。

极大理想、分裂环上的多项式环的因式分解。唯一因子分解环的多项式扩张、环的直和、中国剩余定理、模、子模、模同态、商模、正合列、模的第一同构定理、循环模、直积与直和、自由模、环的整元素。

4， 主理想环上的有限生成模、Noether环，Neother归纳原理、Noether环上的有限模，Hilbert定理与主理想定理。

4。Noether环的幂零根与Jacobson根。

Artin模、Neother模、Krull定理、模的同构定理、投射模、内射模、模的张量积。

5， 域的扩张、有限扩张域，代数扩张、代数封闭。超越扩张、分裂域、Kronecker定理、可分多项式、有限域扩张、有限域的子域、有限域的自同构、Mobius反演公式、

分圆多项式。

6。有限扩张环，完全闭合子环。

7。正规环，有限扩张商域上的完全闭合正规Noether环。

8。有限生成代数，正规化的Noether引理，有限生成代数的有限扩张域与Jacobson根的同态。

6， 代数闭域、域扩张的自同构、Galois群、Artin引理、Galois扩张、Galois理论主定理、尺规做图问题、三等分角问题、倍立方问题、分圆扩张、不可约性判别法、Brauer定理、Dedekind定理、Artin定理、正规基。

9。有限生成代数的超越次数。

10。仿射代数蔟，Hilbert零点定理，代数蔟的直积。

7， 循环扩张、交换扩张、可解扩张、范数和迹、Speiser定理、Artin-Speiser定理、方程可用根式解的判别法、表示、表示空间、表示模。

8， 酉表示、Maschke定理、多面体群、Schur定理、特征标、对称群的表示、Young图、Young表、不可约表示、交换群的表示、特征标群、Frobenius互反定理。

11。Zarisky拓扑，代数蔟的不可约性判别准则。不可约流形上直积的不可约性。

12。不可约代数流形的维数，直积与子流形的维数。

13。多项式域的扩张，他的存在性与唯一性，有限域。

14。Galois扩张，可分多项式的分裂域的Galois扩张理论，三次多项式的Galois群，一般多项式，分圆多项式与有限域。

15。Galois对应，代数方程的可解性。

9， SU(2)群和SU(3)群的表示、表示的张量积、特征标环、有限群中的刚性与有理性、结合代数、商代数、中心单代数、Wedderburn-Artin定理、可除代数、Wedderburn定理、代数的线性表示、表示的不可约与完全可约，完全不可约表示的子空间的不变性，Burnside定理。

10， 矩阵Lie群、矩阵紧Lie群、矩阵Lie群的同态与同构、特殊线性群的极分解、Lie群、Lie代数、Lie代数的表示。

17。紧致线性群：完全不可约与不变轨道的分割，不变量的Hilbert定理。

18。代数闭域上的有限维结合代数的结构，线性表示的不可约性。

19。有限群的不可约线性表示的维数定理，特征标与矩阵元的正交性。

20。四元数代数及其与SO3和SO4群的关系，广义四元数代数。

21。Euclid平面与空间上的运动的有限群。

22。晶体群，Biberbach定理，晶体群的分类。

23。一维与二维上同调群，晶体群的抽象结构的Biberbach定理与Zassenhaus定理。

Groups &Abstract Algebra

1，A.I.Kostrikin，代数学引论，科学出版社，1977。

2，A.I.Kostrikin，代数学引论，物理数学文献出版社，2000。

3，E.B.Vinberg，代数学教程，法克斯特里亚出版社，2001。

4，A.G.Kurosh，高等代数教程，科学出版社，1971。

5，A.I.Kostrikin，代数学习题集，物理数学文献出版社，2001。

代数学

二年级，第一学期

参考书：

1，A.I.Kostrikin，代数学引论，科学出版社，1977。

2，A.I.Kostrikin，代数学引论，物理数学文献出版社，2000。

3，E.B.Vinberg，代数学教程，法克斯特里亚出版社，2001。

4，A.I.Kostrikin，代数学习题集，物理数学文献出版社，2001。

代数学的附加章节

第三学年全年课程

参考书目：

1，E.B.Vinberg，代数学教程，法克特里亚出版社，2002。

2，M.Atiyah、I.McDonald，Introduction To Commutative Algebra，Addison-Wesley，1969。

3，S.Lang，Algebra，Springer，2005。

Basic Algebra I&II, 2nd Edition by N. Jacobson 
Algebra by Serge Lang 
Dummit & Foote “Abstract Algebra” Wiley
Hungerford “Abstract Algebra: An Introduction” Brooks/Cole

Friedberg “Linear Algebra” 4th ed. Prentice Hall
Axler “Linear Algebra Done Right” 2nd ed. Springer-Verlag
Hoffman & Kunz , Linear Algebra

《抽象代数（Schaum’s题解精萃影印版）》 

A.I. Kostrikin，Introduction to algebra，Springer-Verlag

N.Jacobson “Basic Algebra I,II” 

N. Jacobson “Lectures on Abstract Algebra”(GTM.30,31,32) (中译本:抽象代数学,共三卷,理图里有) 

Abstract Algebra Dummit：

Algebra Hungerford：

Algebra M,Artin：

a first course in abstract algebra by rotman。

Advanced Modern Algebra by Rotman：

Algebra：a graduate course by Isaacs：

《近世代数概论》Garrett Birkhoff;Saunders Mac Lane

Robinson “A course in the theory of Groups”(GTM 80) 

10.E.Artin “伽罗华理论” 

Edwards “Galois Theory”(GTM 101) 

丁石孙,聂灵沼 “代数学引论” 

徐诚浩 “抽象代数–方法导引” 

莫宗坚《代数学》(上,下)” 北京大学出版社

熊全淹《近世代数》武汉大学出版社

库洛什 “群论” 

冯克勤《近世代数引论》中国科学技术大学出版社

聂灵沼《代数学引论》高等教育出版社

《代数学》范德瓦尔登

《代数学引论》柯斯特利金

《近世代数基础》张禾瑞

《抽象代数基础》丘维声

《代数学引论》聂灵沼，丁石孙

《抽象代数》（1、2）赵春来等著

《近世代数》杨子胥

《近世代数习题解》杨子胥

此外还有一套《离散数学习题集》里面有

《抽象代数分册》张立昂

《抽象代数学》姚幕生（复旦）

《近世代数基础》刘绍学

《近世代数引论》章璞，李尚志，冯克勤（中科大）

《抽象代数》盛德成（西安）

《整数与多项式》冯克勤余红兵著高等教育出版社

冯克勤，李尚志，查建国，章璞，《近世代数引论》

刘绍学，《近世代数基础》

205《近世代数引论》冯克勤

206《代数学》(上,下) 莫宗坚

207 《近世代数》熊全淹

208《近世代数》盛德成

209《代数学引论》丁石孙，聂灵沼

【习题集】

210《抽象代数–方法导引》徐诚浩

【提高】

211《Algebra》S.Lang

212《伽罗华理论》E.Artin

Algebra -1

Syllabuses on Algebra and Number Theory

Linear Algebra

Abstract vector spaces; subspaces; dimension; matrices and linear transformations; matrix algebras and groups; determinants and traces; eigenvectors and eigenvalues, characteristic and minimal polynomials; diagonalization and triangularization of operators; invariant subspaces and canonical forms; inner products and orthogonal bases; reduction of quadratic forms; hermitian and unitary operators, bilinear forms; dual spaces; adjoints. tensor products and tensor algebras;

Integers and polynomials

Integers, Euclidean algorithm, unique decomposition; congruence and the Chinese Remainder theorem; Quadratic reciprocity ; Indeterminate Equations. Polynomials, Euclidean algorithm, uniqueness decomposition, zeros; The fundamental theorem of algebra; Polynomials of integer coefficients, the Gauss lemma and the Eisenstein criterion; Polynomials of several variables, homogenous and symmetric polynomials, the fundamental theorem of symmetric polynomials.

Group

Groups and homomorphisms, Sylow theorem, finitely generated abelian groups. Examples: permutation groups, cyclic groups, dihedral groups, matrix groups, simple groups, Jordan-Holder theorem, linear groups (GL(n, F) and its subgroups), p-groups, solvable and nilpotent groups, group extensions, semi-direct products, free groups, amalgamated products and group presentations.

Ring

Basic properties of rings, units, ideals, homomorphisms, quotient rings, prime and maximal ideals, fields of fractions, Euclidean domains, principal ideal domains and unique factorization domains, polynomial and power series rings, Chinese Remainder Theorem, local rings and localization, Nakayama’s lemma, chain conditions and Noetherian rings, Hilbert basis theorem, Artin rings, integral ring extensions, Nullstellensatz, Dedekind domains,algebraic sets, Spec(A).

Module

Modules and algebra Free and projective; tensor products; irreducible modules and Schur’s lemma; semisimple, simple and primitive rings; density and Wederburn theorems; the structure of finitely generated modules over principal ideal domains, with application to abelian groups and canonical forms; categories and functors; complexes, injective modues, cohomology; Tor and Ext.

Field

Field extensions, algebraic extensions, transcendence bases; cyclic and cyclotomic extensions; solvability of polynomial equations; finite fields; separable and inseparable extensions; Galois theory, norms and traces, cyclic extensions, Galois theory of number fields, transcendence degree, function fields.

Group representation

Irreducible representations, Schur’s lemma, characters, Schur orthogonality, character tables, semisimple group rings, induced representations, Frobenius reciprocity, tensor products, symmetric and exterior powers, complex, real, and rational representations.

Lie Algebra

Basic concepts, semisimple Lie algebras, root systems, isomorphism and conjugacy theorems, representation theory.

Combinatorics (TBA)

References:

1. Strang, Linear algebra, Academic Press.

2. I.M. Gelfand, Linear Algebra

3. Feng Keqin of the integers and polynomials Yu Hongbing with higher education press

4. Jacobson, Nathan Basic algebra. I. Second edition. W. H. Freeman and Company, New York, 1985. xviii+499 pp.

5. Jacobson, Nathan Basic algebra. II. Second edition. W. H. Freeman and Company, New York, 1989. xviii+686 pp.

6. S. Lang, Algebra, Addison-Wesley

7. Feng Keqin, Li Shangzhi, Cha Jianguo, Zhang Pu, an introduction to the modern algebra

8. Liu Shaoxue, the Foundation of modern algebra

9. J. P. Serre, Linear representations of finite groups

10.     J. P. Serre: Complex semisimple Lie algebra and their representations

11.     J. Humphreys: Introduction to Lie algebra and representation theory, GTM 009.

12.     W. Fulton, Representation theory, a First Course, GTM 129. 

1 , Brief history, algebra linear equations, matrix, stepped into linear equations and matrices,Gauss elimination method, low-order determinant, collection and mapping, binary relations, equivalence relations, business map, a partially ordered set.

2 , Mathematical induction, replacement, replacement of the loop structure, replacement of symbols, skew-symmetric functions, basic concepts of number theory, the fundamental theorem of arithmetic.

3 , Vector and scalar, linear combination, linear correlation and linear independence and the linear independence of the line, the fundamental lemma of the linearly independent, bases and dimensions and the base with the rank of linear equations, matrix of rank, the rank of the matrix compatibility study of linear equations, Linear equations by solvability criteria, linear mapping, linear transformation, linear function inverse matrix, matrix, matrix operations, the equivalent class, fundamental system of solutions of linear equations. The solution space of linear equations.

3 。 Replacement on a finite set, parity of the permutation, Abelian groups, product of permutation is decomposed into rotation.

4 , The determinant of square matrices, As to the volume of the determinant, Determinant of a basic nature, minor, minors, the determinant of a start.

5 , Non-degenerate type of judgement, the determinant criterion of zero matrices by row (column), with matrices, linear equations of theCramer law, jiabianzi law, as

Determinant of a linear specification antisymmetric function. Theorem of rank of a matrix, determinant fandemengde.

5 。 Operation and characteristics of the matrix, inverse matrix, methods of calculating the proposed matrix, fundamental transformation, product of the basic transformation of matrix and fundamental matrices.

6 。 Basic algebra: groups, rings, fields and residue class rings, domains feature.

6 , Two Yuan operation, and half group, and Yao half group, and group, and child group, and cycle group, and group of isomorphism, andCayley theorem, and group of with State and since with State, and ring, and with more than class, and remaining class ring, and ring of with State, and whole ring, and domain, and domain of isomorphism and since isomorphism, and domain of features, and pigment domain, and plural domain, and plural of geometry said, primitive root, and plural of geometry, and make than. Plural form of trigonometry and algebra.

7 , Univariate polynomial rings over fields, Monadic polynomial ring, a ring of multivariate polynomials, Polynomial decomposition product of the non-trivial polynomial potential and uniqueness, The only factorial rings, Largest Convention of two polynomials, Ring the least common multiple and greatest common, central elements of coprime, divisibility of the judgment, Euclid’s algorithm,Euclid Ring, irreducible polynomial and primitive polynomial, Gauss Lemmas, Eisentein Criterion.

8 。 Division of the field of integers and polynomials, the Reals and complex field of irreducible polynomials.

9 。 The roots of polynomials, derivatives and their properties, standard decomposition of roots of polynomials, polynomial root scope, Descartes ‘ theorem, calculate the number of real roots of a polynomial method of Sturm und drang.

8 , Integral field of fractions, rational function field, the most simple fractions,Bezout theorem, polynomial function loop, interpolation polynomial,Lagrange polynomial Newton Interpolation, polynomial rings of differential method,Vieta formula, symmetric and skew-symmetric functions,Wilson theorem.

9 , Symmetric more items type ring, and more said items type of basic theorem, and pending coefficient method, and, power and, andNewton formula, and more items type of discriminant type, and knot type, and plural domain of algebra closed, and algebra basic theorem, and calculation more items type real roots number of Strum theorem, and more items type root of approximate algorithm, and The rational roots of polynomials with integer coefficients.

11 。 Rational fraction field, rational real type is decomposed into partial fractions, and real and complex fields.

12 。 Multivariate polynomial ring, symmetric polynomials, his fundamental symmetric polynomials, Victor theorem.

13 。 Cyclic group, his subgroups, cyclic group isomorphism, cluster decomposition theorem of Lagrange.

14 。 The discriminant of the polynomial, the discriminant of the polynomial relation with their roots.

10 , General fields of linear spaces, subspaces, linear, linear independent vector group of rank, base and dimension, the transition between different matrices, linear space isomorphism, subspace, dimension theorem, direct and, space, commercial space, linear functions, independent of the dual space, linear discriminant method.

11 , Linear maps, matrix representation of linear mapping, such as nuclear, linear operator, linear algebras, minimal polynomials, matrices are similar, the determinant and trace of a linear operator.

12 , Invariant subspace, eigenvalues and eigenvectors, characteristic polynomial, characteristic subspaces, geometry, number and algebra and diagonalizable operators to distinguish law, the existence of invariant subspaces, conjugate linear operators and commercial operators.

Algebra -2

1 , Categories, functors and theHamilton-Cayley theorem, theJordan standard form, root space, cyclic subspace, matrix, matrix rational canonical form.

2 , Matrix polynomials, polynomial of matrix elementary transformation, polynomial matrix,Smith standard, determinant factor, invariant factors, primary group, feature matrix and Jordan standard form similar to the real, real square matrices.

3 , Multiple linear mapping, and double linear type, and matrix of consistency transform, and double linear type of rank, and left Foundation, and symmetric double linear type and oblique symmetric double linear type, and two times type, and two times type of specification type, and of two times type for specification type of method, and real two times type, and inertia theorem, and definite two times type and definite matrix, andJacobi method, andSylvesterTheorem, skew-symmetric quadratic norm type, Pfaff Type.

4 , Euclid spaces, inner product, the standard orthogonal basis,Gram-Schmidt orthogonalization process,Euclid space isomorphism, orthogonal matrices, orthogonal groups, and symplectic space and symplectic, unitary, Symplectic operator space,Hermite , Unitary matrices, the unitary group, normed linear spaces, according to the mode of convergence, absolute convergence.

5 , Within product space Shang of linear operator, and of two times type for spindle form, and put two a two times type while into specification type, and insurance from operator of specification form, and very decomposition, and singular value decomposition, andSchur theorem, andWitt expansion theorem, and complex structure, and complex of linear space, and real of linear space, and real of linear operator, and complex of operator, and Least squares, spheres, weighting orthogonal polynomials.

6 , Norm of a linear operator, a linear group of one-parameter subgroups, the spectral RADIUS, affine space, affine mapping, affine space isomorphism, affine subspace, affine affine coordinate system, isomorphism,Euclid metrics,Gramdeterminant, to the volume.

7 , Affine group,Euclid space group of motions, distance transformation groups, convex sets,Minkowski space, pseudo-Euclidean space,Lorenz Group, an affine space of quadratic function, quadratic function to standardize the type,Euclid Space of quadratic functions.

8 , Quadric, quadric surface Center, standardization of quadric surface in affine space, quadratic surfaces

The classification, Euclid Quadric surface in space, the projective plane, higher dimensional projective space, homogeneous coordinates and projective geometry, affine geometry, algebraic and projective groups, than with the weight ratio of quadric surface in projective space classification, linear and projective quadric surfaces intersect.

9 , Concepts of tensor, tensor coordinates, Zhang Liangji, tensor of convolution, symmetric and skew-symmetric tensor, tensor space, the exterior algebra.

1 。 Group, subgroup isomorphic to group, group order, cyclic groups, generated set of symmetric groups, linear groups and alternating groups, Cayley’s theorem.

10 , Normal subgroups, representatives of left cosets and right cosets, Yuan,Lagrange theorem, cyclic groups of structures, the role of groups in the collection, Group action, rail, stabilizers and orbit, stable subgroup, the conjugacy classes of the group. Normalizer, transitive groups and homogeneous spaces.

11 , Typical groups, homomorphism, the Quaternion algebra, permutation groups, symmetry.

12 , Quotient groups, and Group homomorphisms, Basic theorems, isomorphism theorems, a homomorphism

3 。 Ring ideal, quotient ring, ring too theorem, the splitting field of the polynomial, finite fields.

Transposition group Commutator direct product of groups, group and Semidirect products, generators, free groups, solvable groups, Limited R- Solvability of groups and matrix groups and alternating groups of finite order. Simple groups.

1 , Zassenhaus lemma and theJordan-Holder theorem, the operator group automorphism, endomorphism rings, groups and theSylow theorems, characteristic subgroup,Abel Group Free Abelian group, Finitely generated Abel Group, Frobenius-Stickelberger Theorem, finite Abel The fundamental theorem of group. Structure theorem for finitely generated Abelian groups.

7 。 Domain algebra, Florida binisi theorem on the real numbers, algebras.

8 。 The representation theory of finite groups initially, characters, representation and full representation, Mashooq full representation theorem for finite groups, regular representation of a group theory, regularity of the character, the dimension of the irreducible representation.

Algebra -3

2 , A well-ordered set,Zorn ‘s lemma, the axiom of choice and the morphisms, natural transformations, ideals and quotient ring of a ring, with the State the fundamental theorem, ring isomorphism theorems, ideal for operations, localization, a prime ideal.

3 , Gauss integral, a principal ideal ring,

1 。 Principal ideal rings and Euclid loop, principal ideal rings and Euclid limited model on the ring structure.

Great ideals, Factorization of polynomial rings over a Division ring. Unique factorization of polynomial expansion of the ring, ring, Chinese remainder theorem, die, die, die homomorphism, quotient module, right column, die the first isomorphism theorem, loop mode, direct product and direct sum, free mode, loops the entire element.

4 , Principal ideal rings of finitely generated modules, andNoether ring,Neother induction principle,Noether limited modules over rings,Hilbert Theorem and the principal ideal theorem.

4 。 Noether of nilpotent ring with Jacobson .

Artin Mold, Neother Mold, Krull Isomorphism theorem, theorem, die cast mold, injective tensor products, die.

5 , Domain expansion, finite extension fields, algebraic expansion, algebraically closed. Beyond expansion, splitting field,Kronecker theorem, can be divided into a number of children of the expansion, finite fields, finite fields, finite fields isomorphic andMobius inversion formula,

Cyclotomic polynomial.

6 。 Finite extension rings, completely closed loops.

7 。 Regular rings and finite extension field on a closed formal Noether ring.

8 。 Finitely generated algebra and regularization of Noether ‘s lemma, a finitely generated algebra of finite extension field Jacobson homomorphism.

6 , Algebraically closed fields, automorphisms of field extensions,Galois groups,Artin lemmas,Galois expansion,Galoistheory the master theorem, ruler diagram problems Trisection of angle problems, Doubling the cube problem, expanding circle, not about sexual discrimination law,Brauer theorem,Dedekind theorem,Artin theorem, normal basis.

9 。 Finitely generated algebra over the number.

10 。 Affine algebraic nest,Hilbert nullstellensatz, the direct product of algebraic mounting.

7 , Cycle solutions Exchange expansion, expansion, expansion, norm and trace andSpeiser theorem, theArtin-Speisertheorem, radical solutions to the equation of discriminant method, presentation, presentation spaces, said die.

8 , Unitary representation,Maschke theorem, polyhedral group,Schur theorem, character, representation of the symmetric group,Young diagrams,Young table, irreducible representations, Characters of group representation, Exchange groups,Frobenius reciprocity theorem.

11 。 Zarisky topology, algebraic cocooning irreducibility criterion. An irreducible manifold of a direct product of irreducible.

12 。 Dimension of an irreducible algebraic manifold, direct product and submanifolds of dimension.

13 。 Domain of polynomial expansion, his existence and uniqueness of a finite field.

14 。 Galois expansion can be divided into the splitting field of the polynomial’s Galois expansion theory, cubic polynomial’s Galois Group, generic polynomials, cyclotomic polynomial and finite fields.

15 。 Galois correspondence, solvability of algebraic equations.

9 , SU (2) and the SU (3) Group representations, tensor product, feature ring, rigid and rational in a finite group, the Union, Central simple algebra, algebra, algebraWedderburn-Artin theorem, division algebra, Wedderburn theorem, algebraic linear representation, irreducible and completely reducible, completely irreducible representations of subspaces invariant,Burnsidetheorem.

10 , Matrix Lie groups, matrix compact Lie Group, the matrix Lie Group homomorphism and isomorphism, special linear group of polar decomposition,Lie Group andLie Algebra,Lie algebra representations.

17 。 Compact linear group: altogether about as much as the same orbit segment invariant Hilbert theorem.

18 。 Finite-dimensional associative algebra over an algebraically closed field structure, linear representations of irreducibility.

19 。 Irreducible linear representations of finite dimension theorem, character and matrix element of orthogonality.

20 。 Quaternion algebra and its SO3 and SO4 groups and generalized Quaternion algebra.

21 。 Euclid the motion of plane and space of finite groups.

22 。 Crystallographic groups,Biberbach theorem, the classification of crystallographic groups.

23 。 One and two dimensional cohomology groups, Crystallographic groups the abstract structure of Biberbach theorem and the Zassenhaus theorem.

Groups &Abstract Algebra

1 , A.I.Kostrikin , Introduction to algebra, science press, 1977 。

2 , A.I.Kostrikin , Introduction to algebra, physics and mathematics literature Publishing House, 2000 。

3 , E.B.Vinberg , Algebra tutorials, Ashfaq, Asia Publishing House, 2001 。

4 , A.G.Kurosh , Advanced algebra tutorials, science press, 1971 。

5 , A.I.Kostrikin And algebra problem sets, and physical-mathematical literature Publishing House, 2001 。

Algebra

Second year first semester

Reference books:

1 , A.I.Kostrikin , Introduction to algebra, science press, 1977 。

2 , A.I.Kostrikin , Introduction to algebra, physics and mathematics literature Publishing House, 2000 。

3 , E.B.Vinberg , Algebra tutorials, Ashfaq, Asia Publishing House, 2001 。

4 , A.I.Kostrikin And algebra problem sets, and physical-mathematical literature Publishing House, 2001 。

Additional sections of the algebra

Third year year courses

Bibliography:

1 , E.B.Vinberg , Algebra tutorials, faketeliya Publishing House, 2002 。

2 , M.Atiyah 、 I.McDonald , Introduction To Commutative Algebra , Addison-Wesley , 1969 。

3 , S.Lang , Algebra , Springer , 2005 。

Basic Algebra I&II, 2nd Edition by N. Jacobson 
Algebra by Serge Lang 
Dummit & Foote “Abstract Algebra” Wiley
Hungerford “Abstract Algebra: An Introduction” Brooks/Cole

Friedberg “Linear Algebra” 4th ed. Prentice Hall
Axler “Linear Algebra Done Right” 2nd ed. Springer-Verlag
Hoffman & Kunz , Linear Algebra

Abstract algebra ( Schaum’s Solution highlights print version)

A.I. Kostrikin , Introduction to algebra , Springer-Verlag

N.Jacobson “Basic Algebra I,II”

N.Jacobson”LecturesonAbstractAlgebra”(GTM.30,31,32)( translation : abstract algebra , three volumes , )

Abstract Algebra Dummit :

Algebra Hungerford :

Algebra M,Artin :

a first course in abstract algebra by rotman 。

Advanced Modern Algebra by Rotman :

Algebra : a graduate course by Isaacs :

The introduction of modern algebra Garrett Birkhoff; Saunders Mac Lane

Robinson “A course in the theory of Groups”(GTM 80)

10.E.Artin ” Galois theory”

Edwards “Galois Theory”(GTM 101)

Ding shisun , Nie Ling Moor ” An introduction to algebra”

Xu Chenghao ” Abstract algebra — Method guidance”

Mo Zongjian algebra ( Shang , Xia )” Peking University Press

Xiong Quanyan modern algebra, Wuhan University Press

Kulos ” Group theory”

Feng Keqin The introduction to modern algebra University of science and technology of China Press

Nie Ling moor the introduction to algebra, higher education press

The algebra of van der waerden

Introduction to the algebra of kesitelijin

Rui Zhang Wo, the Foundation of modern algebra

The sound based on high-dimensional abstract algebra

The introduction to algebra Nie Ling Moor, Ding shisun

Of the abstract algebra ( 1 、 2 ) Zhao chunlai waiting

Of the modern algebra Yang Zixu

Of the solution to modern algebra learning Yang Zixu

In addition there is a set of discrete mathematics problem set

Of the abstract algebra fascicle Zhang Liang

The abstract algebra Yao (Fudan University)

The Foundation of modern algebra Liu Shaoxue

The introduction to modern algebra Zhang Pu, Li Shangzhi, Feng Keqin (Science)

The abstract algebra Cheng Decheng (XI an)

Feng Keqin Yu Hongbing the integers and polynomials with higher education press

Feng Keqin, Li Shangzhi, Cha Jianguo, Zhang Pu, an introduction to the modern algebra

Liu Shaoxue, of the Foundation of modern algebra

205 The modern algebra an introduction to Feng Keqin

206 The algebra ( Shang , Xia ) Mo Zongjian

207 The modern algebra Xiong Quanyan

208 The modern algebra Cheng Decheng

209 Introduction to the algebra of Ding shisun, Nie Ling Moor

“Onward”

210 The abstract algebra — Xu Chenghao method guidance

“Increase”

211 《 Algebra 》 S.Lang

212 Of the Galois theory E.Artin