Of course some quintic and higher degree equations can be solved by radicals. For instance the quintic x - 1 = 0 has as solutions the five fifth roots of unity, four of which are non-real. The conditions under which a polynomial equation is solvable by radicals were first discovered by Evariste Galois (1811-32). In order to do this he had to introduce the concept of a group. Here and there in the algebraic research of the 18th century there occurred concepts which began to point to group theory. After some initial work done by Joseph de Lagrange (1736 - 1813) and Paolo Ruffini (1765-1822), Galois succeeded in taking the decisive step. In the night before a duel in which he died, he wrote the fundamental principles of group theory in a letter to a friend. It was many years later before the mathematicians of the day recognised that these ideas on the transformation group of the roots of a polynomial equation were the beginnings of modern algebra.

Since the freeing of algebra from the necessity of geometric interpretation, the letters in an algebraic expression had continued to be considered as standing for numbers, and the symbols between the letters were for the operations of ordinary arithmetic. Now in this first example of a group, the objects being treated algebraically were permutations (of roots) and the operation for combining them was not arithmetical, but had some of the familiar properties of arithmetical operations, such as associativity. Gradually other sets of mathematical objects, with certain operations were recognised to have similar properties, and it became of interest to study the algebraic structure of such systems, independently of the type of the underlying mathematical objects.

Some algebraic properties gave rise to structures more fruitful for study than others. The properties which distinguished a group were, for a given set S with binary operation * :

- For any a and b in S, a*b is also in S

- For any a, b, c in S, (a*(b*c)) and ((a*b)*c) are equal

- There is a member e in S such that a*e = e*a = a for all a in S

- For each a in S there is an a' in S such that a'*a = a*a' = e.

With further research it became apparent which sets of axioms would give rise to algebraic structures worthy of study. Those algebraic structures which proved most rewarding were given names such as "group", "ring", "field", "semigroup", "module" and there is even a type of algebraic structure called an "algebra". The study of each of these is the study of abstract algebra for its own sake, irrespective of what the letters "stand for".

A further outcome of the freeing of algebra from arithmetic was the study of algebraic structures which had the properties not possessed by the algebra of numbers. Examples were the algebra of quaternions, introduced by William Rowan Hamilton (1805-65) in 1844, and the algebra of the "Laws of Thought" (now called Boolean Algebra), introduced by George Boole (1815-64) in 1854.

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