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

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 moreWe develop a very straightforward way to compute the Fourier transform of $\exp(-cx^2)$, in the sense that any contour integration and the calculus of residues are not required at all. This cool approach enables us to think about these elementary concepts much deeper, so I highly recommend to study this approach as long as you are familiar with ODE of first order.

Read moreWe offer a detailed proof of the Riemann mapping theorem, which states that every proper simply connected region is conformally equivalent to the open unit disc.

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