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.
2025/12/29
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Boolean ring and algebraic numbers
In this post, we study the Boolean ring and see how it can be used in algebraic number theory.
2025/10/13
阅读更多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 $X^p-X-\alpha$.
2025/5/16
阅读更多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.
2025/5/10
阅读更多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.
2023/11/11
阅读更多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.
2023/10/21
阅读更多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.
2023/9/18
阅读更多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.
2023/8/12
阅读更多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.
2023/7/17
阅读更多Chinese Remainder Theorem in Several Scenarios of Ring Theory
We study the Chinese remainder theorem in various contexts and abstract levels.
2023/5/27
阅读更多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.
2023/4/6
阅读更多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.
2023/3/20
阅读更多Vague Convergence in Measure-theoretic Probability Theory - Equivalent Conditions
We give an introduction to vague convergence and see several equivalent conditions of it.
2023/2/13
阅读更多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.
2022/12/23
阅读更多The Haar Measure on the Field of p-Adic Numbers
In this post we study the canonical Haar measure on $Q_p$, and give a explicit definition just as the Lebesgue measure.
2022/12/20
阅读更多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.
2022/12/4
阅读更多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.
2022/12/2
阅读更多A Step-by-step of the Analytic Continuation of the Riemann Zeta Function
We compute the analytic continuation of the Riemann Zeta function and after that the reader will realise that asserting $1+2+\dots=-\frac{1}{12}$ without enough caution is not a good idea.
2022/11/24
阅读更多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.
2022/9/22
阅读更多Calculus on Fields - Heights of Polynomials, Mahler's Measure and Northcott's Theorem
We study the height of polynomials and derive some important tools.
2022/9/4
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