A blog mainly on graduate mathematics.
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1970-01-01 08:00:00
In this post we determine $SL_2(\mathbb{F}_3)$ using Sylow theory and linear algebra.
1970-01-01 08:00:00
We show that a separable extension is solvable by radical iff it is solvable, i.e. it has a Galois closure with solvable Galois group. The proof is done in a general setting.
1970-01-01 08:00:00
We show that the range of a non-constant entire function's range cannot be a twice-punctured plane.
1970-01-01 08:00:00
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.
1970-01-01 08:00:00
This post collects the top 5 most popular posts according to Google search console.
1970-01-01 08:00:00
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.
1970-01-01 08:00:00
We study the Chinese remainder theorem in various contexts and abstract levels.
1970-01-01 08:00:00
In this post we study projective representations of $SO(3)$, although we will make more use of $SU(2)$. At the end of this post we reach the conclusion that one will think about polynomials with odd or even terms. Projective representations have its own significance in physics although the room of this post is too small to contain it. Nevertheless, the reader is invited to use linear algebra much more extensively with a taste of modern physics in this post.
1970-01-01 08:00:00
In this post we deliver the basic computation of the quadratic reciprocity law and see its importance in algebraic number theory.
1970-01-01 08:00:00
We give an introduction to vague convergence and see several equivalent conditions of it.
1970-01-01 08:00:00
In this post we show that the Pontryagin dual group of $\mathbb{Q}_p$ is isomorphic to itself.
1970-01-01 08:00:00
In this post we study the canonical Haar measure on $Q_p$, and give a explicit definition just as the Lebesgue measure.
1970-01-01 08:00:00
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.
1970-01-01 08:00:00
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.
1970-01-01 08:00:00
We compute the analytic continuation of the Riemann Zeta function and after that the reader will realise that asserting $1+2+\dots=-\frac{1}{12}$ without enough caution is not a good idea.
1970-01-01 08:00:00
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.
1970-01-01 08:00:00
We study the height of polynomials and derive some important tools.
1970-01-01 08:00:00
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 $\mathbb{Q}_p$.
1970-01-01 08:00:00
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.
1970-01-01 08:00:00
We give a quick look at the Segre embedding and try to use that in a fundamental tool of Diophantine Geometry - heights.