Groups and Fields: The Two Fundamental Notions of Algebra

Abstract

Algebra is the mathematical discipline that arose from the problem of solving equations. If one starts with the integers ${\mathbb Z}$, one knows that every equation $a+x=b$, where a and b are integers, has a unique solution. However, the equation $ax=b$ does not necessarily have a solution in ${\mathbb Z}$, or it might have infinitely many solutions (take $a=b=0$).