The term "algebra" is derived from the Arabic word "al-jabr"  which, in mathematical writing, meant the addition of equal terms to  both sides of an equation in order to eliminate negative terms, or the multiplication of both sides of an equation by equal terms in order to eliminate fractions.  And indeed, up until the early nineteenth century, algebra was essentially the science of solving equations.

Perhaps the most famous problem from this early period of algebra was  the attempt to solve various classes of equations "by radicals."  The classical Greeks knew how to solve quadratic (2nd degree) equations "by radicals" --  this is nothing more than the well  known quadratic formula.  In the sixteenth century similar methods were devised for the cubic (3rd degree) and quartic (4th degree) equations.  But the quintic (5th degree) equation proved difficult to handle -- in fact, for almost three centuries, mathematicians tried in vain to devise a method to solve a quintic equation "by radicals."  Then, in 1824, a young man named Niels Abel gave an ingenious demonstration that indeed it was impossible to solve a general quintic equation in this way.

However, an even more amazing and far reaching solution to the  problem of solving equations "by radicals" was obtained by Evariste Galois.  In 1829, with apparently no knowledge of Abel's result, Galois succeeded in finding conditions for a polynomial equation of any degree that would determine whether or not the equation was solvable ``by radicals.''  A corollary of Galois' result was that, for n greater than or equal to 5, there is no solution "by radicals" for the general nth degree equation.

The importance of Galois's theorem, great as it is, is probably less than the importance of the concepts and methods employed in stating and proving the result:  they changed the nature of algebra forever.  To each equation, Galois associated a special object which today is called the Galois group of the equation.  Galois' theorem is then stated as:

A "group" is a collection of objects with a binary operation (two elements in the group will produce a third) that behaves a lot like multiplication.  ``Solvability'' of a group is a property that a group may or may not have -- although we will not define solvability at this time, it suffices to observe that any specific group can be tested for this property far more easily that a specific equation can be tested for solvability "by radicals."

A group is a specific type of algebraic structure:  an arbitrary set with one or more operations defined on it, the operations obeying some collection of (algebraic) rules.  Because of the pioneering work of Galois, the character of algebra switched from developing methods for solving equations to the systematic study of the properties of algebraic structures.

MCS313 Home