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.
Segre Embedding And 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.
The Calculus of Fields - Absolute Values
Irreducible Representations of SO(3) and the Laplacian
In this post, irreducible representations of $SO(3)$ are studied, with much more extensive applications of linear algebra.
Irreducible Representations of SO(3) and the Laplacian
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.
Study Irreducible Representations of SU(2) Using Fourier Series
The Fourier Transform of exp(-cx^2) and Its Convolution
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.
The Fourier Transform of exp(-cx^2) and Its Convolution
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.
A Detailed Proof of the Riemann Mapping Theorem
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 3 - Polynomials of Prime Degree and Pairs of Nonreal Roots
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 2 - Cubic Extensions
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.
Examples in Galois Theory 1 - Complex Field is Algebraically Closed
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 Group Algebra of A Finite Group and Maschke's Theorem