Calculus on Fields - Heights of Polynomials, Mahler's Measure and Northcott's Theorem
We study the height of polynomials and derive some important tools.
Calculus on Fields - Heights of Polynomials, Mahler's Measure and Northcott's Theorem
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 $\mathbb{Q}_p$.
Hensel's Lemma - A Fair Application of Newton's Method and 'Double Induction'
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.
Irreducible Representations of GL_2(F_q)
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
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