$SU(2)$ has a lot of interesting mathematical and physical properties. In this post we study its irreducible representations in a mathematician's way.

We develop two almost 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. The first 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.

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.

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.

We study the Galois group of a cubic polynomial over a field with characteristic not equal to 2 and 3.

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.

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.

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.

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.

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