We show that a separable extension is solvable by radical iff it is solvable, i.e. it has a Galois closure with solvable Galois group. The proof is done in a general setting.

In this post we study cyclotomic polynomials in field theory and deduce some baisc properties of it. We will also use it to solve some problems in field theory.

We study the height of polynomials and derive some important tools.

In this episode we focus on the rational field. What can we know about the Galois group of an irreducible polynomial with prime degree? There is a method by counting the number of nonreal roots. From this, we obtain an algorithm to compute the Galois group.

We study the Galois group of a cubic polynomial over a field with characteristic not equal to 2 and 3.

We try to prove the fundamental theorem of algebra, that the complex field is algebraically closed, using as little analysis as possible. In other words, the following proof will be *almost* algebraic.

In this post, we study the concept of character, what it is about in abstract harmonic analysis and how to use it Galois theory.