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.
