What does it mean for a group to be Galois?

Definition (Galois Group): If F is the splitting field of a polynomial p(x) then G is called the Galois group of the polynomial p(x), usually written \mathrm{Gal}(p). So, taking the polynomial p(x)=x^2-2, we have G=\mathrm{Gal}(p)=\{f,g\} where f(a+b\sqrt{2})=a-b\sqrt{2} and g(x)=x.

What does Galois theory do?

In a word, Galois Theory uncovers a relationship between the structure of groups and the structure of fields. It then uses this relationship to describe how the roots of a polynomial relate to one another.

Is Galois group always Abelian?

In abstract algebra, an abelian extension is a Galois extension whose Galois group is abelian. When the Galois group is also cyclic, the extension is also called a cyclic extension. A cyclotomic extension, under either definition, is always abelian.

Is Galois group cyclic?

When the Galois group is also cyclic, the extension is also called a cyclic extension. Every finite extension of a finite field is a cyclic extension. Class field theory provides detailed information about the abelian extensions of number fields, function fields of algebraic curves over finite fields, and local fields.

How many elements are in Galois?

Let G be the Galois group of a field with nine elements over its subfield with three elements.

What is a Galois group?

A Galois group tells you how you can shuffle around the roots of some polynomial in ways that preserve nice algebraic properties. To actually get a feeling of what this means, there is less value in coming up with a simple (or complicated) explanation than actually seeing lots of examples. So, let’s do that.

What does Galois theory depend on for symmetries?

Clearly Galois Theory depends a lot on the concept of symmetries: group theory, that is. I am assuming that you have done a course on group theory, and have met the concept of normal subgroups. The great news is that Galois theory gives us a better intuition of what normal subrgoups are. I delve into that in another post. Getting into the details.

What is the Galois group of a polynomial?

That is, the Galois group in this case consists of four functions: i d, σ, σ 2, σ 3 (in this context, σ 2 ( x) = σ ( σ ( x)) ). In each of the preceding examples, the size of the Galois group was always equal to the degree of the polynomial.