Elements of Algebra

John Stillwell

Let’s not beat the bush; algebra is a branch of abstract mathematics. On the other hand, algebra also represents practical mathematics in its most ideal and unadulterated form. Abstraction is not pursuing its own sake; rather, it is being pursued efficacy, power, and insight. After 2,000 years of other forms of mathematics failing, algebra developed from the quest to answer actual, physical issues in geometry. Algebra was the first type of mathematics to be successful in this endeavor.

This was accomplished by revealing the mathematical structure underlying geometry and giving tools with which to investigate that structure. This is a typical example of how algebra can be used; it is the best and most pure form of application since it elucidates the mathematical structures that are both the most fundamental and the most general. This book was written to foster a proper appreciation of algebra by demonstrating the application of abstraction to concrete issues, specifically the traditional challenges of building using a straightedge and compass.

These difficulties date back to Euclid’s time when geometry and number theory was of the utmost importance; nonetheless, they were not resolved until the 19th century, when abstract algebra was developed. As is common knowledge, algebra is the key to unifying not only geometry and number theory but also the majority of the subfields that make up mathematics. When one has a historical background of the topic, which I also intend to share, something like this is not shocking.