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
The Structure of SL_2(F_3) as a Semidirect Product
In this post we determine $SL_2(\mathbb{F}_3)$ using Sylow theory and linear algebra.
The Structure of SL_2(F_3) as a Semidirect Product
The abc Theorem of Polynomials
In this post we show the Mason-Stothers theorem, the so-called $abc$ theorem for polynomials, and derive Fermat's Last theorem and Davenport's inequality for polynomials. These three theorems correspond to the $abc$ conjecture, Fermat's Last Theorem and Hall's conjecture in number theory.
The abc Theorem of Polynomials
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 Group Algebra of A Finite Group and Maschke's Theorem
In this post we give several forms of Masher's theorem by studying group algebra, which eventually becomes a study of semisimple rings. One can consider this post a chaotic evil introduction to representation theory or something.
The Group Algebra of A Finite Group and Maschke's Theorem
Dedekind Domain and Properties in an Elementary Approach
Dedekind Domain and Properties in an Elementary Approach
Why Does a Vector Space Have a Basis (Module Theory)
Why Does a Vector Space Have a Basis (Module Theory)
Study Vector Bundle in a Relatively Harder Way - Tangent Bundle
Study Vector Bundle in a Relatively Harder Way - Tangent Bundle
Study Vector Bundle in a Relatively Harder Way - Definition
Study Vector Bundle in a Relatively Harder Way - Definition