- Education and Science
Galois and Group Theory
Galois Theory a Branch of Group Threory (Maths)
Galois Theory is a branch of Group Theory (Maths) which was invented by the talented young Mathematician Evariste Galois who's life was tragically cut short (he was shot in a duel) just days after first publishing his work at the age of just twenty in 1832. His theory was used to prove the centuries old problem of the insolvability of fifth-degree algebraic equations.
In 1984 (as a precocious teenager) I wrote a thesis about Galois and Group Theory and this article is taken from that work although I have not reproduced all of my mathematics here. I have made some reading suggestions for further research or study into this fascinating subject.
Not very useful, but quite interesting...
Taken from a thesis written by me in 1984 (as a precocious teenager)
The need for Group Theory arose in connection with a problem which is of little importance. The problem was the solution of algebraic equations. Linear and quadratic equations were solved centuries B.C. but equations of third and fourth degree were not solved until about 1550 A.D. and no progress in this respect has been made since. No one has succeeded in solving equations of fifth degree. It was proved by means of group theory (Galois Theory) that it was in fact an impossible task. This branch of algebra has no direct application to any purpose. There is rarely a situation in which an equation of greater than fourth degree needs to be solved algebraically because it is easier to draw a graph or to use an iterative method such as Newton Raphson, especially when a computer is used. Thus Galois Theory has no real application in mathematics, but is interesting as a study of more complex group theory.
What is a Group?
Definition of a Group
A Group is a system of elements under a single valued binary operation which satisfies the following four conditions:Closure: If two elements are combined by a given operation the result must be an element in the system (e.g. If one integer is added to another integer the result is also an integer)Identity Element: The system must contain an identity (or neutral) element, which when combined with any other element in the system leaves it unchanged. (e.g. Adding zero to any integer)Inverse Element: Every element must have an inverse element such that if an element is combined with its inverse the result is the identity element. (e.g. The inverse of 3 under addition is -3)The Associative Law must hold: e.g. a + (b + c) = (a + b) + c
These are the only conditions that must be obeyed for the system to be a group. If other conditions are obeyed the group gains additional characteristics ( e.g. Abelian Groups: commutative isomorphic groups) A group does not have to be finite and its elements need not be numbers, but could be translations, rotations or matrices. The operation could be almost anything, not just arithmetic functions.
The "order" of a group is the number of elements in the group and a group may have several sub-groups of different orders.
A Cyclic Group is a group in which the elements of each row and column follow the same order.
Lagrange's Theorem states that the order of a subgroup of a finite group and the periods (orders) of all of the elements are factors of the order of the group.
For more about Groups and the use of Group Theory please see the recommended reading.
The Short Tragic Life of Galois - Galois Theory
Evariste Galois was born in the village of Bourg-La-Reine, just outside Paris, on 25 October, 1811.
He went to the Lycee of Louis-Lle-Grand in Paris (at the age of twelve) where his work was described as mediocre and he was described as peculiar, but he had tremendous mathematical talent. So he took the entrance examination for L'Ecole Polytechnique and failed it twice.
He sent some of his outstanding work to Fourier and Cauchy, who lost the manuscript.
He was expelled from school and imprisoned for being a revolutionist.
He was 'framed' for duel, so he quickly wrote down his theories and was shot the next day. He died eighteen days later on 31 May 1832 (at the age of 20)
Where Did The Mathematician Galois Live?
Applications of Galois Theory
Solutions of Equations
Throughout history people have been solving algebraic equations: over two thousand years ago equations of the first order were solved:
ax + b = c can be solved unless a=0 and b unequal to 0 as x = -b/a
Ancient Babylonians were able to solve equations of the second degree many centuries B.C.
ax2 + bx + c = 0 has roots of x = -b +/- SQRT(b2 - 4ac) / 2a
Equations of third and fourth degree were not solved until the sixteenth century and mathematicians believed that some day equations of higher degree would also be solved, but it was not unil the 19th Century that, by means of the theory of groups (Galois Theory) that is was found to be impossible.
Whether a problem can or cannot be solved depends on the conditions imposed upon the solution.
e.g. x + 5 =3 can be solved if negative numbers are permitted and 2x + 3 = 10 can be solved id fractions are allowed.
A field is a set of numbers such that the sum, difference, product and quotient (division by zero being ruled out)_ of any two of them are also in the set.
An algebraic expression may be reducible or irreducible depending on the field in which the solutions are to found.
so, all complex numbers form a field, as do real numbers, rational numbers, but integers do not form a field (since the quotient of two integers may not be an integer)
An equation of fifth degree appeared to be a kind of mathematical atom, which cannot be broken down any further.
An example of an unatomic problem is x^6 = a can be broken down into two parts: y^2 = a and x^3 = y^2 where y^2 = a and x^3 = y^2 are both atomic (neither can be reduced to two simple problems)
The Group of an Equation
Galois showed that every algebraic equation is connected with a group and by examining this group it is possible to determine whether the equation is solvable, or at least whether it can be broken down into simpler parts.
For more about how to use Galois Theory please see the reading suggestions