# Properties of Cyclotomic Polynomials

In this post we study cyclotomic polynomials in field theory and deduce some baisc properties of it. We will also use it to solve some problems in field theory.

# Calculus on Fields - Heights of Polynomials, Mahler's Measure and Northcott's Theorem

We study the height of polynomials and derive some important tools.

# 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 $\mathbb{Q}_p$.

# 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.

# Segre Embedding And Heights

We give a quick look at the Segre embedding and try to use that in a fundamental tool of Diophantine Geometry - heights.

# 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.