Modern Algebra

Donovan D Chang

Modern Algebra

What is modern algebra? Another name for modern algebra is abstract algebra. Before we think about modern or abstract algebra, let us first think about algebra. Following is the definition from the Oxford Dictionary [3]:

“The part of mathematics in which letters and other general symbols
are used to represent numbers and quantities in formulae and equations”

The above definition provided by Oxford most closely reflects the traits of elementary algebra and linear algebra. In elementary algebra, we learn

four basic arithmetic operations: addition, subtraction, multiplication, and division ( $+,-,\times,\div$ );
the concepts of \emph{known} and \emph{unknown} variables, where the latter, the unknown, is most often denoted by $x$ ;
how to simplify mathematical expressions,
e.g. $x\times x+x+x=x^2+2x=x(x+2)$ ; and
how to set up equations and inequalities, and some basic techniques how to solve them.

Linear algebra primarily deals with systems of linear equations, whether the solutions are real or complex numbers, referring to the system consistent, when there exists a solution, and inconsistent otherwise. As we deal with multiple variables, it becomes cumbersome to write the same variables over and over again, and thus we introduce the notion of vector. Also, almost always is the case a linear system is comprised of multiple equations, and these can be conveniently handled in a matrix form, either as a stack of vertical (column) vectors, or horizontal (row) vectors.

Finally we turn our focus to abstract algebra. Indeed, as implied from the name, it is algebra of abstract nature. To be specific, instead of the results of binary operations themselves and finding a solution to given equations or inequalities, our focus is on the process and a pattern, or collectively an algebraic structure.

Following is an illustrative example. Try to spot the similarity in the process.

0 in addition
For any $x\in\mathbb{C}$ ,

$x+0=0+x=x$
1 in multiplication
For any $y\in\mathbb{C}$ ,

$y\cdot 1=1\cdot y=y$
Identity element $e\in A$ of a binary operation $*$ defined on $A$ .
For any $z\in A$ ,

$z*e=e*z=z$

Here is another example.

$-x$ of $x$ in addition
For any $x\in\mathbb{C}$ ,

$x+(-x)=(-x)+x=0$
$\frac{1}{y}$ of $y$ in multiplication
For any $y\in\mathbb{C}\setminus \left\lbrace 0 \right\rbrace$ ,

$y\cdot \frac{1}{y}=\frac{1}{y}\cdot y=1$
Inverse $z'$ of $z$ , given a binary operation $*$ defined on $A$ .
For any $z\in A$ , there exists $z'\in A$ such that

$z*z'=z'*z=e$

Did you spot a pattern, or similarity? Let us begin our exploration to formally define what we have just observed.

Modern Algebra

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