ABSTRACT
It is one of the fascinating features of mathematics that quite often the introduction of a new concept allows a problem to be solved that seemed intractable before. That is what happened when Evariste Galois (1811–1832) set about tackling the century-old problem of solving polynomial equations. If p is a polynomial of degree 2 then the solutions of the equation p(x) = 0 are given by the well-known quadratic formula. Mathematicians tried to find analogous formulas for polynomials of higher degree. It turned out, however, that such formulas do not always exist. It was Galois’ brilliant idea to associate with each polynomial p a group G (which we now call the Galois group of p) such that there is a formula for the solutions of the equation p(x) = 0 if and only if the group G has a certain property. This property can be formulated in purely group-theoretical terms; it is called solvability because of its close connection to the solvability of polynomial equations. Thus the problem of solving polynomial equations was completely translated into a purely group-theoretical problem. We will discuss Galois’ theory in the field theory chapter in Volume II of this book. In this section we introduce and discuss solvable groups and also the closely related class of nilpotent groups. It will turn out that these groups share many properties with abelian groups.
