Algebra 

Algebra is a discipline of mathematics dealing with sets (see set theory), which are structured by one or more binary operations. While studying these so-called algebraic structures (i.e. groups, rings, fields, modules, and vector spaces), algebra provides means to find solutions of equations and systems of equations formulated inside these structures.

Theoretical minimum (in a nutshell)

You should be acquainted with set theory, especially the set operations and basics about functions.

Concepts you will learn in this part of BookofProofs

• What are groups, which are their properties and applications?
• What are rings and which are the different types of rings and their properties?
• What are fields and how they improve the solvability of equations in comparison to rings?
• What are vector spaces, matrices, determinants and how these and other concepts of linear algebra help to solve systems of linear equations?
• What are modules?
• How the Galois theory helps to form new fields out of existing ones and what solvability of equations has to do with geometry?

| | | | created: 2014-02-20 20:27:42 | modified: 2014-04-30 21:29:02 | by: bookofproofs

1.Algebraic Structures - Overview 

2.Motivation: Common Concepts Of Algebra: Substructures and Morphisms 

3.Group Theory 

4.Algebraic Number Theory and Ring Theory 

5.Field Extensions 

6.Finite Fields 

7.Galois Theory 

8.Linear Algebra 

9.Constructions with Ruler and Compass 

10.Universal Algebras 

11.Categories 

12.Solving Strategies and Sample Solutions to Problems in Algebra