On the Boyd–Deninger polynomial x+1/x+y+1/y+1, pt. I - The curve
In this post we study the Boyd-Deninger polynomial P(x,y)=x+1/x+y+1/y+1. In particular, we are interested in the elliptic curve that is defined by it.
On the Boyd–Deninger polynomial x+1/x+y+1/y+1, pt. I - The curve
Boolean ring and algebraic numbers
In this post, we study the Boolean ring and see how it can be used in algebraic number theory.
Boolean ring and algebraic numbers
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
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
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.
Picard's Little Theorem and 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.
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