Monday, September 23, 2013

Abstract Algebra Part: Learning outcomes

We are going to learn Abstract Algebra about this topic. Firs look, what are we are going to learn or our learning course content.

1.1 Defining Groups
1.2 Defining Rings
1.3 Examples
1.4 First Concepts
1.5 Ideals
1.6 Homomorphisms
1.7 Products
2 Commutative Rings
2.1 Maximal Ideals
2.2 Prime Ideals
2.3 Radical Ideals
2.4 Noetherian Rings
2.5 Unique Factorisation Domains
2.6 Principal Ideal Domains
2.7 Lasker-Noether Decomposition
2.8 Finite Rings
2.9 Localisation
2.10 Local Rings
2.11 Dedekind Domains
3 Modules
3.1 Defining Modules
3.2 First Concepts
3.3 Free Modules
3.4 Homomorphisms
3.5 Rank of Modules
3.6 Length of Modules
3.7 Localisation of Modules
 4 Linear Algebra
4.1 Matices
4.2 Elementary Matrices
4.3 Linear Equations
4.4 Determinants
4.5 Rank of Matrices
4.6 Canonical Forms
5 Structure Theorems
5.1 Asociated Primes
5.2 Primary Decomposition
5.3 The Theorem of Prufer
6 Polynomial Rings
6.1 Monomial Orders
6.2 Graded Algebras
6.3 Defining Polynomials
6.4 The Standard Cases
6.5 Roots of Polynomials
6.6 Derivation of Polynomials
6.7 Algebraic Properties
6.8 Grobner Bases
6.9 Algorithms
7 Polynomials in One Variable 209
7.1 Interpolation
7.2 Irreducibility Tests
7.3 Symmetric Polynomials
7.4 The Resultant
7.5 The Discriminant
7.6 Polynomials of Low Degree
7.7 Polynomials of High Degree
7.8 Fundamental Theorem of Algebra
8 Group Theory
9 Multilinear Algebra
9.1 Multilinear Maps
9.2 Duality Theory
9.3 Tensor Product of Modules
9.4 Tensor Product of Algebras
9.5 Tensor Product of Maps
9.6 Di erentials
10 Categorial Algebra
10.1 Sets and Classes
10.2 Categories and Functors
10.3 Examples
10.4 Product
10.5 Abelian Categories
10.6 Homology
11 Ring Extensions
12 Galois Theory
13 Graded Rings
14 Valuations
15 Proofs - Fundamentals
16 Proofs - Rings and Modules 246
17 Proofs - Commutative Algebra 300
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