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