We show that the range of a non-constant entire function's range cannot be a twice-punctured plane.

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.

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.

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.

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.

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.

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

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.