Last modified 21 May 2008
One of the fundamental problems in algebra is to find solutions to polynomial equations. Algebraic Geometry is a branch of mathematics studying the solutions to families of polynomials involving many variables. Some modern crytographic techniques are based on solutions to elliptic functions.
Galois Theory refers to finding roots of a polynomial in a single variable, for example
In modern terminology, it associates to a polynomial the field generated by all its roots and considers the group of automorphisms of this field. The question of whether a polynomial is solvable by radicals or not is then replaced by the question of whether the group is a solvable group or not. In fact, this is the origin of the term solvable group. One should remark that Thompson has just been awarded the Abel prize for his work on finite groups, including the celebrated Feit-Thompson Theorem, which proves that every finite group with an odd number of elements is solvable.
Galois Theory is seen as one of the starting points of modern algebra, where symmetry plays a central role. In fact, symmetry is prevalent in mathematics, physics (for example super-symmetric string theory) and also art (Girih tilings in Islamic architecture).
Everyone learns in school that the roots of a quadratic equation are given by a formula, although these roots may be complex numbers. In fact, if
then the roots of f are given by
This formula was essentially known to the Babylonians (1600 BC).
A similar formula exists for cubic polynomials. This was discovered by Fontana, but is commonly referred to as Cardano's Formula, since it was first published by him in 1545. Ferrari, a student of Cardano, found a formula for the quartic polynomial. This was also published in Cardano's manuscript.
The formulae for the roots of quadratic, cubic and quartic polynomials involve the usual arithmetic operations of addition, subtraction, multiplication and division, together with taking square roots and cube roots. It was therefore a natural question whether similar formulae existed for all polynomials, if we also allow taking 5-th roots, 7-th roots, and p-th roots for all prime numbers p. (It is enough to consider primes, since every number can be written as a product of primes.) Such expressions are called radical expressions, and we say that a polynomial is solvable by radicals if such a formula exists. Hence all quadratic, cubic and quartic polynomaials are solvable by radicals.
Lagrange later unified the methods for quadratic, cubic and quartic polynomials, and showed that they all depend on finding functions of the roots which are invariant under certain permutations, hence have certain symmetries. He also showed that this approach cannot work for a quintic polynomial.
Extending ideas of Ruffini, Abel proved in 1824 that no single formula can work for all quintic polynomials. Kronecker later published a simple, rigorous proof in 1879.
This didn't quite solve the problem, though, since some polynomials are obviously solvable by radicals. For example
is solvable by radicals, since it has the solution
The possibility therefore still existed that each quintic polynomial had its own special formula.
The answer was found by Galois in 1830. Given a polynomial, he considered the group of all symmetries of its roots, now called its Galois group. He then showed that the polynomial is solvable by radicals if and only if its Galois group has a chain of subgroups such that each subgroup is normal in the next and with cyclic factor group. Such groups are now called solvable groups. Since one can exhibit quintic polynomials having Galois group the symmetric group on 5 letters, which is known not be solvable, for example
one sees that in general, polynomials of degree at least 5 are not solvable by radicals.
A. Hubery
Last modified 21 May 2008