Mathematics. Articles in English (et en français dans le futur).

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

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

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'

Segre Embedding And Heights

We give a quick look at the Segre embedding and try to use that in a fundamental tool of Diophantine Geometry - heights.
Segre Embedding And Heights

The Calculus of Fields - Absolute Values

In this post, we develop a fundamental device: absolute value on an arbitrary field with various points of view.
The Calculus of Fields - Absolute Values