Posted 2023-03-20Updated 2023-03-30Number Theory / Algebraic Number Theory / Elementary Number TheoryThe Quadratic Reciprocity LawIn this post we deliver the basic computation of the quadratic reciprocity law and see its importance in algebraic number theory.Read more
Posted 2022-12-23Updated 2022-12-23Number Theory / p-Adic AnalysisThe Pontryagin Dual group of Q_pIn this post we show that the Pontryagin dual group of $\mathbb{Q}_p$ is isomorphic to itself.Read more
Posted 2022-12-20Updated 2022-12-20Number Theory / p-Adic AnalysisThe Haar Measure on the Field of p-Adic NumbersIn this post we study the canonical Haar measure on $Q_p$, and give a explicit definition just as the Lebesgue measure.Read more
Posted 2022-09-04Updated 2022-09-04Algebra / Galois Theory / Number Theory / Algebraic Number TheoryCalculus on Fields - Heights of Polynomials, Mahler's Measure and Northcott's TheoremWe study the height of polynomials and derive some important tools.Read more
Posted 2022-08-21Updated 2022-12-21Algebra / Number Theory / Algebraic Number Theory / Commutative Algebra / p-adic Analysis / Local FieldsHensel'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 $\mathbb{Q}_p$.Read more
Posted 2022-07-26Updated 2022-08-26Number Theory / Algebraic Number Theory / Geometry / Algebraic Geometry / Diophantine GeometrySegre Embedding And HeightsWe give a quick look at the Segre embedding and try to use that in a fundamental tool of Diophantine Geometry - heights.Read more
Posted 2022-06-30Updated 2022-06-30Algebra / Number Theory / Field Theory / Algebraic Number TheoryThe Calculus of Fields - Absolute ValuesIn this post, we develop a fundamental device: absolute value on an arbitrary field with various points of view.Read more