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 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.
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.
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.
Dedekind Domain and Properties in an Elementary Approach
You can find contents about Dedekind domain (or Dedekind ring) in almost all algebraic number theory books. But many properti...
Why Does a Vector Space Have a Basis (Module Theory)
Module and vector spaceFirst we recall some backgrounds. Suppose $A$ is a ring with multiplicative identity $1_A$. A left mod...
The Grothendienck Group
Free groupLet $A$ be an abelian group. Let $(e_i)_{i \in I}$ be a family of elements of $A$. We say that this family is a bas...
Study Vector Bundle in a Relatively Harder Way - Tangent Bundle
Tangent line and tangent surface as vector spacesWe begin our study by some elementary Calculus. Now we have the function $f(...
Study Vector Bundle in a Relatively Harder Way - Definition
MotivationDirection is a considerable thing. For example take a look at this picture (by David Gunderman): The position of t...
Cauchy sequence in group theory
Recall - Cauchy sequence in analysisBefore we go into group theory, let’s recall how Cauchy sequence is defined in analysis. ...