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.

Read moreThe 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.

Read moreA 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.

Read moreSL(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.

Read moreArtin'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.

Read moreChinese Remainder Theorem in Several Scenarios of Ring Theory

We study the Chinese remainder theorem in various contexts and abstract levels.

Read moreProjective 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.

Read moreEvery 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.

Read moreThe 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.

Read moreProperties 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.

Read moreCalculus on Fields - Heights of Polynomials, Mahler's Measure and Northcott's Theorem

We study the height of polynomials and derive some important tools.

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