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
Hensel's Lemma - A Fair Application of Newton's Method and 'Double Induction'
We prove the celebrated Hensel's lemma using the so-called Newton's method and "double induction", and try to find solutions of polynomials in $\mathbb{Q}_p$.
Hensel's Lemma - A Fair Application of Newton's Method and 'Double Induction'
Irreducible Representations of GL_2(F_q)
In this post we follow the step of Fulton-Harris to classify all irreducible representations of $GL_2(\mathbb{F}_q)$. A character table is added at the end.
Irreducible Representations of GL_2(F_q)