# 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.

# Irreducible Representations of SO(3) and the Laplacian

In this post the irreducible representations of $SO(3)$ are studied, with much more extensive applications of linear algebra.

# 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.

# The Fourier transform of exp(-cx^2)

We develop a very straightforward way to compute the Fourier transform of $\exp(-cx^2)$, in the sense that any contour integration and the calculus of residues are not required at all. This cool approach enables us to think about these elementary concepts much deeper, so I highly recommend to study this approach as long as you are familiar with ODE of first order.

# A Detailed Proof of the Riemann Mapping Theorem

We offer a detailed proof of the Riemann mapping theorem, which states that every proper simply connected region is conformally equivalent to the open unit disc.

# 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.

# 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.

# 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.

# 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.

# 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.