Desvl's blog

Mathematics. Articles in English (et en français dans le futur).

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.

2022-01-18 Algebra Representation Theory Group Algebra Non-Commutative Ring

# Study new content # Serge Lang # Axiom of Choice # Division Ring # J-P Serre

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

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.

2021-12-29 Algebra Commutative Algebra Geometry Algebraic Geometry

# Study new content

The ring of trigonometric polynomials with complex scalars

The ring of real trigonometric polynomials

The ring

Throughout we consider the polynomial ring

$R=\mathbb{R}[\cos{x},\sin{x}].$

This ring has a lot of non-trivial properties which give us a good chance to study commutative ring theory.

2021-12-24 Algebra Commutative Algebra

# Study new content # Noetherian

The ring of real trigonometric polynomials

Characters in Analysis and Algebra

In this post, we study the concept of character, what it is about in abstract harmonic analysis and how to use it Galois theory.

2021-10-10 Analysis Fourier Analysis Algebra Galois Theory

Characters in Analysis and Algebra

Dedekind Domain and Properties in an Elementary Approach

2021-02-07 Algebra Commutative Algebra

# Exercise solution # Serge Lang # Dedekind

Dedekind Domain and Properties in an Elementary Approach

Tensor Product as a Universal Object (Category Theory & Module Theory)

2021-01-19 Algebra Commutative Algebra Category Theory Module Theory

# Universal Property

Tensor Product as a Universal Object (Category Theory & Module Theory)

Why Does a Vector Space Have a Basis (Module Theory)

2021-01-15 Algebra Module Theory

# Serge Lang # Axiom of Choice

Why Does a Vector Space Have a Basis (Module Theory)

Rings of Fractions and Localisation

2021-01-14 Algebra Commutative Algebra

# Atiyah-MacDonald # Universal Property

Rings of Fractions and Localisation

The Grothendienck Group

2021-01-10 Algebra Group Theory

# Serge Lang # Universal Property # Grothendieck

The Grothendienck Group

A long exact sequence of cohomology groups (zig-zag and diagram-chasing)

2020-09-27 Algebra Homological Algebra

# Diagram-chasing # Differential complex

A long exact sequence of cohomology groups (zig-zag and diagram-chasing)