Desvl's blog

Artin-Schreier Extensions

We are interested in a special category of field extensions. Let $K$ be a field of characteristic $p \ne 0$, we want to know the structure of an extension of $K$ of degree $p$. It turns out that there lies the an Artin-Schreier polynomial of the form $X^p-X-\alpha$.

A Separable Extension Is Solvable by Radicals Iff It Is Solvable

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.

Properties of Cyclotomic Polynomials

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.

Calculus on Fields - Heights of Polynomials, Mahler's Measure and Northcott's Theorem

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

Examples in Galois Theory 3 - Polynomials of Prime Degree and Pairs of Nonreal Roots

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.

Examples in Galois Theory 2 - Cubic Extensions

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

Examples in Galois Theory 1 - Complex Field is Algebraically Closed

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.

Characters in Analysis and Algebra

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