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The word 'algebra' is derived form the Arabic words Al Jabr from the title of a book Al Jabr Wa'l Muqabalah which means "restoration by transposing terms from one side of the equation to the other". The book was written by an Arab mathematician Muhammed Ibn Musa Al-Khwarizmi in the 800's in Baghdad. Most readers are probably familiar with algebra from high-school and the solution of linear and quadratic polynomials over the real numbers. Abstract algebra is the study of algebraic methods without the restriction of using the set of real or even complex numbers. It also examines the various structures over which the rules of algebra apply to in general.
Any study of mathematics is much more than just the study of numbers and rules. It is primarily a study in logic and should be read accordingly. A mathematics text cannot just be 'read' like other texts where you are just absorbing a number of facts. Granted, there are certain definitions and notations that must be learned for the sake of communicating ideas, but a mathematics book should be read in such a way that you 'internalize' the material. When a proof is offered, the proof should be reflected upon until the logic is understood. This (and all math books) are subject to typographical errors, mistakes in logic, and omissions. If when analyzing a proof it does not make sense, don't assume that you have misunderstood. It is wise to have other sources on the subject available or work out the proof for yourself.
This document is written in HTML and has taken advantage of some of the features of internet browsers that are not available in normal texts. Whenever possible, links to the definitions of various items and statements of theorems have been provided so the reader can quickly reference them. Also mouseover definitions are included in many places where appropriate and practical. Also links to short biographies of mathematicians mentioned in the text have been provided. These are intended to give the reader an occasional diversion and to provide some background on the motivation that resulted in their theorems and conjectures. Many students of abstract algebra are left wondering why anyone would have developed such ideas. Hopefully the biographies will give some insight into this.
Before we embark on the rest of our study, I feel we should mention a few definitions of words that the reader is probably familiar with, but possibly never knew the definition of
Definition 1: A conjecture is a statement that has not been proven.
Definition 2: A theorem is a statement to which a proof has been given. The proof is a rigorous justification of the veracity of the statement in such a way that it cannot be disputed by anyone who follows the rules of logic and accepts a given set of axioms.
Definition 3: A lemma is a preliminary proven statement which leads to a theorem.
Definition 4: A corollary is a statement of a proven result which follows from a theorem.
No expectations of the reader are made, that is, there are no prerequisites needed to read this material. Some experience and 'mathematical maturity' from a course in calculus would be helpful, but not necessary. This is meant to be a fully contained document that anyone can use to better understand the logic and tools of algebra.
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