Galois and Group Theory

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.

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.

Field Theory

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