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.
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.
Picard's Little Theorem and Twice-Punctured Plane
We show that the range of a non-constant entire function's range cannot be a twice-punctured plane.
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.
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.
Chinese Remainder Theorem in Several Scenarios of Ring Theory
We study the Chinese remainder theorem in various contexts and abstract levels.
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.
The Quadratic Reciprocity Law
In this post we deliver the basic computation of the quadratic reciprocity law and see its importance in algebraic number theory.
Vague Convergence in Measure-theoretic Probability Theory - Equivalent Conditions
We give an introduction to vague convergence and see several equivalent conditions of it.
The Pontryagin Dual group of Q_p
In this post we show that the Pontryagin dual group of $\mathbb{Q}_p$ is isomorphic to itself.