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?
Table of Contents
- Part: Algebraic Structures - Overview
- Part: Group Theory
- Part: Algebraic Number Theory and Ring Theory
- Part: Finite Fields
- Part: Galois Theory
- Part: Linear Algebra
- Part: Constructions with Ruler and Compass
- Part: Ordered Fields and Their Topology
- Part: Solving Strategies and Sample Solutions to Problems in Algebra