# Properties of Cyclotomic Polynomials

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.

Read moreIn 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.

Read moreWe study the height of polynomials and derive some important tools.

Read moreWe prove the celebrated Hensel's lemma using the so-called Newton's method and "double induction", and try to find solutions of polynomials in $\mathbb{Q}_p$.

Read moreIn 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.

Read moreIn this post, we develop a fundamental device: absolute value on an arbitrary field with various points of view.

Read moreIn this post the irreducible representations of $SO(3)$ are studied, with much more extensive applications of linear algebra.

Read more$SU(2)$ has a lot of interesting mathematical and physical properties. In this post we study its irreducible representations in a mathematician's way.

Read moreIn 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.

Read moreWe study the Galois group of a cubic polynomial over a field with characteristic not equal to 2 and 3.

Read moreWe 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.

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