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 $X^p-X-\alpha$.
Artin-Schreier Extensions
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.
A Separable Extension Is Solvable by Radicals Iff It Is Solvable
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.
Properties of Cyclotomic Polynomials
The Calculus of Fields - Absolute Values
In this post, we develop a fundamental device: absolute value on an arbitrary field with various points of view.
The Calculus of Fields - Absolute Values