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 $\mathbb{Q}_p$.

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.

In this post, we develop a fundamental device: absolute value on an arbitrary field with various points of view.

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

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

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.