Picard's Little Theorem and Twice-Punctured Plane
We show that the range of a non-constant entire function's range cannot be a twice-punctured plane.
SL(2,R) As a Topological Space and Topological Group
In this post we show that $SL(2,\mathbb{R})$ can be identified as the inside of a solid torus and see what we can learn from it.
Artin's Theorem of Induced Characters
We give a relatively more detailed proof of Artin's theorem in representation theory of finite groups as well as an example of dihedral group.
Every Regular Local Ring is Cohen-Macaulay
In this post we show that the class of regular local rings (the abstract version of power series rings) is a subclass of Cohen-Macaulay ring.
The abc Theorem of Polynomials
In this post we show the Mason-Stothers theorem, the so-called $abc$ theorem for polynomials, and derive Fermat's Last theorem and Davenport's inequality for polynomials. These three theorems correspond to the $abc$ conjecture, Fermat's Last Theorem and Hall's conjecture in number theory.
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, irreducible representations of $SO(3)$ are studied, with much more extensive applications of linear algebra.