Modern Algebra

Modern, or abstract, algebra is the study of algebraic structures, specifically binary algebraic structures. Binary algebraic structures are those sets that are defined under at least one binary operation that maps an element $(a,b)$ in the set $A\times B$ to an element $c$ in the set $C$. While most other branches of mathematics are focused on mathematical operations themselves, modern algebra focuses on the different sets and their underlying structure [21]. In fact, the discovery of one structural property of some sets helped make this branch what it is today. When we think of multiplication, we usually assume it is commutative. This is understandable as any set of numbers worked with in elementary school, high school, and even most undergraduate math courses, on which multiplication can be applied has the property of commutativity. Consequently, the discovery of sets that possessed multiplication that was not commutative led to much of the development of modern algebra [21].

Many algebraic structures exist, but in this paper, we will only be discussing a few of them, groups, rings, integral domains, and fields. But even the study of these few algebraic structures has many applications. For example, the simplest finite field consisting of only 0 and 1 influenced the formation of the binary system in data communication and coding [21]. This branch also affected number theory. There existed the problem of taking “a finite number of variables” and deciding “which ideals could be generated by at most finitely many polynomials” [21]. In the late 1800s, the German mathematician David Hilbert used the concept of ideals (which we will briefly discuss in connection to rings and integral domains) to solve this number theory problem [21]. Another question of number theory is which whole numbers can be written as the sum of squares. This problem can be solved by considering how factorization works in the ring of complex numbers [21]. Modern algebra has also influenced geometry. By using rings, it is possible to take some geometric problems, turn them into algebraic ones, solve them, and then transfer the results back to geometry [21].

Modern Algebra is unique in that it focuses on the structure of sets, and it is amazing to see the structural similarities between algebraic structures that at first glance look quite different. In this paper, we will begin by discussing groups. These algebraic structures form much of the foundation for what is studied in Modern Algebra. We will also discuss a special type of subgroup, known as a cyclic subgroup, that is found in every single group. We will then look at isomorphisms which tell us whether two groups have the same structural properties. Next, we will introduce cosets and factor groups, which are made up of cosets. We will also introduce a couple of theorems related to cosets and isomorphisms before moving onto rings

Rings are what we get when we take a special type of group and add the binary operation of multiplication that must only satisfy the distributive property. We will look at a special type of ring known as an integral domain and a specific type of integral domain known as a field. We will finish our discussion of modern algebra by looking at rings of polynomials and determining whether a polynomial is reducible over a given field or not. We will also introduce a couple of theorems and a corollary that will help us prove irreducibility. So, let’s begin our look into the unique and interesting branch of mathematics known as Modern Algebra.