Equivalent Conditions of Regular Local Rings of Dimension 1
In this post we collect and prove (as detailed as possible) the equivalent conditions of being a Regular local ring of dimension 1.
Equivalent Conditions of Regular Local Rings of Dimension 1
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
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
SL(2,R) As a Topological Space and Topological Group
In this post we show that $SL(2,\mathbb{R})$ can be identified as the inside of a solid torus and see what we can learn from it.
SL(2,R) As a Topological Space and Topological Group
Artin's Theorem of Induced Characters
We give a relatively more detailed proof of Artin's theorem in representation theory of finite groups as well as an example of dihedral group.
Artin's Theorem of Induced Characters
Chinese Remainder Theorem in Several Scenarios of Ring Theory
We study the Chinese remainder theorem in various contexts and abstract levels.
Chinese Remainder Theorem in Several Scenarios of Ring Theory
Projective Representations of SO(3)
In this post we study projective representations of $SO(3)$, although we will make more use of $SU(2)$. At the end of this post we reach the conclusion that one will think about polynomials with odd or even terms. Projective representations have its own significance in physics although the room of this post is too small to contain it. Nevertheless, the reader is invited to use linear algebra much more extensively with a taste of modern physics in this post.
Projective Representations of SO(3)
Every Regular Local Ring is Cohen-Macaulay
In this post we show that the class of regular local rings (the abstract version of power series rings) is a subclass of Cohen-Macaulay ring.
Every Regular Local Ring is Cohen-Macaulay
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