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Posted 2022-12-23Updated 2022-12-23Number Theory / p-Adic Analysis

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.
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Posted 2022-12-20Updated 2022-12-20Number Theory / p-Adic Analysis

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.
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Posted 2022-09-04Updated 2022-09-04Algebra / Number Theory / Galois Theory / Algebraic Number Theory

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

We study the height of polynomials and derive some important tools.
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Posted 2022-08-21Updated 2022-12-21Algebra / Commutative Algebra / Number Theory / Algebraic Number Theory / p-adic Analysis / Local Fields

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 $\mathbb{Q}_p$.
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Posted 2022-07-26Updated 2022-08-26Number Theory / Algebraic Number Theory / Geometry / Algebraic Geometry / Diophantine Geometry

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.
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Posted 2022-06-30Updated 2022-06-30Algebra / Field Theory / Number Theory / Algebraic Number Theory

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.
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Desvl

Desvl

A blog mainly on graduate mathematics.

A Noetherian local ring

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