# Every Regular Local Ring is Cohen-Macaulay

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.

Read moreIn 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.

Read moreIn 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.

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, 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 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.

Read moreThis post is a continuation of a previous post about the ring of trigonometric polynomials over the real field. Now we have jumped into the complex field, and the extension is not a trivial matter.

Read more