Contents
Main Discussion
Galois Group
Now that groups and fields have been described, it is time to define the Galois group.
For starters, define a group G. Referring back to field extensions, if there exists an extension F of Q, then there exists a grouping of automorphisms of Q onto F. Let the group G be the container of these automorphisms. In general, this basic definition is referred to as the Galois group of the field extension. However, if the field F is actually the splitting field of a polynomial, then it can be called the Galois group of that polynomial.
If the Galois group is a grouping of automorphisms of a field, then how can one know it is a group? What is its operation? A Galois group makes use of function composition as its operation, f * g, where f and g are members of the Galois group.
The notation of a Galois group involves using Gal(K), where K is an input. This can be denoted Gal(F/Q) if one is describing the Galois group of a field extension, and Gal(P) for a polynomial if P is a polynomial.
So, what's the purpose of such an abstractly defined structure? The answer to this question involves what a Galois group is capable of doing. For instance, if a Galois group is found for a polynomial p(x), and one proves that this Galois group is soluble, then the polynomial has radical roots. This is an important idea for proving that general equations do not exist for polynomials of a certain degree as seen later in the Abel-Ruffini Theorem.
Fundamental Theorem of Galois Theory
Abel-Ruffini Theorem
Polynomials of degree 1 to 4 have been studied and formulae to find their solutions have been described. However, polynomials of degree 5 and greater have been shown to not be solvable in the general sense. As a disclaimer, this does not mean that all polynomials of degree 5 or greater have no solution; a quintic equation x^5 - 1= 0 obviously has a solution of one. This result can be arrived at using Galois Theory. The entire proof of the general solvability of a polynomial of degree 5 or higher spans many pages of a typical print size and would be absolutely unreachable to an undergraduate student of reasonable mathematical knowledge. Thus, this page is dedicated to the application of Galois Groups to proving this long standing question of mathematics, by looking at the major key points of the proof. The result of this proof is known as the Abel-Ruffini theorem.
In order to approach proving the general case of a polynomial of degree 5 or higher as unsolvable, one must prove that the Galois Group of a polynomial p(x) of degree 5 or higher is an unsolvable group. The best way to do this in the general case is to construct a Galois Group using the general solution of a polynomial. The Abel-Ruffini theorem not only makes use of Galois Groups but also symmetric groups as discussed earlier.
