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Posted 2022-08-21Updated 2023-07-08Algebra / Number Theory / Algebraic Number Theory / Commutative Algebra / 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 2022-08-12Updated 2023-07-08Algebra / Representation Theory / Linear Algebra / Finite Groups / Finite Fields

Irreducible Representations of GL_2(F_q)

In this post we follow the step of Fulton-Harris to classify all irreducible representations of $GL_2(\mathbb{F}_q)$. A character table is added at the end.
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Posted 2022-06-30Updated 2023-07-08Algebra / Number Theory / Field 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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Posted 2022-06-16Updated 2023-07-08Algebra / Representation Theory / Linear Algebra / Lie Groups

Irreducible Representations of SO(3) and the Laplacian

In this post, irreducible representations of $SO(3)$ are studied, with much more extensive applications of linear algebra.
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Posted 2022-05-08Updated 2023-07-08Algebra / Analysis / Representation Theory / Linear Algebra / Lie Groups / Fourier Series

Study Irreducible Representations of SU(2) Using Fourier Series

$SU(2)$ has a lot of interesting mathematical and physical properties. In this post we study its irreducible representations in a mathematician's way.
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Posted 2022-03-29Updated 2023-07-08Algebra / Galois Theory / Group Theory

Examples in Galois Theory 3 - Polynomials of Prime Degree and Pairs of Nonreal Roots

In this episode we focus on the rational field. What can we know about the Galois group of an irreducible polynomial with prime degree? There is a method by counting the number of nonreal roots. From this, we obtain an algorithm to compute the Galois group.
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Posted 2022-03-27Updated 2023-10-21Algebra / Galois Theory / Group Theory

Examples in Galois Theory 2 - Cubic Extensions

We study the Galois group of a cubic polynomial over a field with characteristic not equal to 2 and 3.
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Posted 2022-03-16Updated 2023-10-21Algebra / Galois Theory / Group Theory

Examples in Galois Theory 1 - Complex Field is Algebraically Closed

We try to prove the fundamental theorem of algebra, that the complex field is algebraically closed, using as little analysis as possible. In other words, the following proof will be *almost* algebraic.
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Posted 2022-01-18Updated 2023-07-08Algebra / Representation Theory / Group Algebra / Non-Commutative Ring

The Group Algebra of A Finite Group and Maschke's Theorem

In this post we give several forms of Masher's theorem by studying group algebra, which eventually becomes a study of semisimple rings. One can consider this post a chaotic evil introduction to representation theory or something.
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Posted 2021-12-29Updated 2023-07-08Algebra / Commutative Algebra / Geometry / Algebraic Geometry

The ring of trigonometric polynomials with complex scalars

This post is a continuation of a previous post about the ring of trigonometric polynomials over the real field. Now we have jumped into the complex field, and the extension is not a trivial matter.
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Desvl

Desvl

A blog mainly on graduate mathematics.

A Noetherian local ring

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