The Structure of SL_2(F_3) as a Semidirect Product
In this post we determine $SL_2(\mathbb{F}_3)$ using Sylow theory and linear algebra.
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.
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 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 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.
The Grothendienck Group
Free groupLet $A$ be an abelian group. Let $(e_i)_{i \in I}$ be a family of elements of $A$. We say that this family is a bas...
Cauchy sequence in group theory
Recall - Cauchy sequence in analysisBefore we go into group theory, let’s recall how Cauchy sequence is defined in analysis. ...