Fields

A field can be described as a set of numbers that allow for the operations of addition, subtraction, multiplication, and division (excluding zero).

There exists a notation for fields and what is called a field extension. Here is an example of a field and an extension of that field:

1. Q - the rational numbers

2. Q[√3] - Q extended by √3

Q is simply the set of all rational numbers. However, Q[√3] is the set of all numbers of the form a + b√3 (a and b are of the set of rational numbers). If one lets X = Q[√3] , then the field extension can be written as X/Q, or Q[√3]/Q. X is the new field that extends Q, while Q is a member set of the larger field of X.

In order to understand the definition of the Galois group, an understanding of a splitting field is required. A splitting field is basically the smallest field that also includes the radical solutions of a polynomial. So, if p(x) = x^4 - 4 = 0, the field that includes the solutions of the polynomial would be Q[√2]. This also works for a polynomial of the form x^6 -7x^3 + 10. The splitting field of this polynomial would be Q[∛2, ∛5].

One more idea that ought to be introduced before tackling the Galois group is the idea of an automorphism. An automorphism takes the form of a function that can be acted on the field and has the property of invertibility. It also has the unique property that f(x + y) = f(x) + f(y), f(ax) = f(a)*f(x), f(1/x) = 1/f(x).

Expanding upon this definition, there is a unique automorphism for an extended field. If K/F is a field extension, then if one takes the F automorphism of K, the resulting automorphism f also follows the property f(x) = x for all members of F.

Now that both groups and fields have been discussed, one may embark on learning about the Galois group


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Abstract algebra continues the conceptual developments of linear algebra, on an even grander scale.

Dr. Paul Garrett