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

# 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.

# 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 $\mathbb{Q}_p$.

# 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.

# 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.