In this post we deliver the basic computation of the quadratic reciprocity law and see its importance in algebraic number theory.

# Vague Convergence in Measure-theoretic Probability Theory - Equivalent Conditions

We give an introduction to vague convergence and see several equivalent conditions of it.

# 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 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.

# 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.

# 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.

# 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.

# 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.

# Calculus on Fields - Heights of Polynomials, Mahler's Measure and Northcott's Theorem

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

# 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 $\mathbb{Q}_p$.