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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.
2023-03-20 Number Theory Algebraic Number Theory Elementary Number Theory
# Serre
The Quadratic Reciprocity Law

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

We study the height of polynomials and derive some important tools.
2022-09-04 Algebra Number Theory Galois Theory Algebraic Number Theory
# Heights# Diophantus
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 $\mathbb{Q}_p$.
2022-08-21 Algebra Commutative Algebra Number Theory Algebraic Number Theory p-adic Analysis Local Fields
# Exercise solution# Study new content# Hensel
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.
2022-07-26 Number Theory Geometry Algebraic Number Theory Algebraic Geometry Diophantine Geometry
# Study new content# Height
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.
2022-06-30 Algebra Field Theory Number Theory Algebraic Number Theory
# Study new content
The Calculus of Fields - Absolute Values