24 History

In our last chapter we discussed the relations and structures for abstract mathematics as well as the study of space for real numbers. Unlike analysis and abstract mathematics where we were introduced to the various rules and theorems for manipulating sets, sequences, and logical propositions, modern or abstract algebra concerns the study of the algebraic structures themselves and less on the manipulation of them. Structures under abstract algebra include, fields, groups, rings, and vector spaces, all of which we will cover in this section.

Much of what is covered in Abstract or Modern Algebra is the structural study of sets under algebra. The course itself follows a very organized pattern starting small, with our fundamental understanding of sets, before introducing the basics of binary algebraic structures and the axioms for groups, subgroups, and isomorphic structures. Once we have a strong understanding of the space and formations of sets we move on to study rings which involve multiple binary operators on a set and then further on to understanding the qualifications for a set to be a field.

It would be improper if we did not cover some of the history behind abstract algebra and who created which concepts that shaped modern algebra into what it is today. When we step back and look at abstract algebra, so many of the structures we are about to discuss only exist because of the coupling of specific axioms, or rules, that must be satisfied in order for a set to be classified under these certain structural families. We can trace back these axioms to Euclid.

Born in the mid 4th century, Euclid was a mathematician who studied a very primitive version of what we recognize today as geometry. He used a foundation of theorems, proofs, and previously defined mathematical terms to create Euclidean geometry which is still utilized today [33]. The birth of axioms came when Euclid provided 5 general structural requirements for geometric shapes. By utilizing axioms or rules to defend whether a structure was of a certain category or not Euclid was able to simplify geometry. Instead of studying and testing every structure people could simply test a structure against the required axioms to prove its classification.

Moving on from the foundation to the discovery of concepts within Modern Algebra, we first look at Carl Gauss. Born in 1777, Carl Gauss has been considered one of the key founders of mathematics. He shaped the understanding of complex numbers using a combination of real and imaginary numbers using . Later on in his studies Gauss was able to provide a proof to support The Fundamental Theorem of Algebra. The theorem supports many ideas but one of the most important to the study of abstract algebra is that it supports the idea that the field of complex numbers is closed [33]. We are familiar with open sets and closed sets from our previous chapter on Analysis. In 1801, Gauss published his book “Disquisitiones Arithmeticae”. The book is considered one of the most influential mathematics books and contains the foundation for modern number theory which we will explore later on in this chapter.

Much of the work done to establish groups in abstract algebra came from a combination of work provided by Carl Gauss, Leopold Kronecker, and Heinrich Weber. Each one contributed properties and axioms for group structures. The man who contributed the most to solving algebraic equations directly was Joseph-Louis Lagrange who created the Lagrange theorem relating the order of a group to the order of a subgroup[33]. Fields were later established by contributing research from Ernst Steinitz who published many works on field theory as well as established the properties for fields including algebraic closure [33].

There are numerous mathematicians who all made ground breaking discoveries and published countless works on the properties of algebraic structures but we can not cover them all. We can, however, spend the time going over their discoveries, taking a look at some examples to better understand the definitions they tirelessly worked for. We begin our section on Modern Algebra with the all encompassing area of groups.