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)
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.
The Quadratic Reciprocity Law
Vague Convergence in Measure-theoretic Probability Theory - Equivalent Conditions
We give an introduction to vague convergence and see several equivalent conditions of it.
Vague Convergence in Measure-theoretic Probability Theory - Equivalent Conditions
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.
The Pontryagin Dual group of Q_p
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.
The Haar Measure on the Field of p-Adic Numbers
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
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.
A Step-by-step of the Analytic Continuation of the Riemann Zeta Function
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
Calculus on Fields - Heights of Polynomials, Mahler's Measure and Northcott's Theorem
We study the height of polynomials and derive some important tools.
Calculus on Fields - Heights of Polynomials, Mahler's Measure and Northcott's Theorem