This document is a seminar note on singular homology. Before reading this article, make sure you have known the Chapter 1-3 of An Introduction to Algebraic Topology, written b…
Here is my note of the category theory, which is the basic of modern mathematics. While it's very useful, it's too hard and abstract to learn. I want to record my unde…
Sylow 定理是群论中的重要定理, 使我们更了解有限群的结构. 但是其证明过程在很多书上比较繁琐而且让人摸不着头脑, 本文整理了Advanced Modern Algebra 这本书中的证明方法, 其更为清晰且自然. ...
蛇形引理是同调代数中的一个重要引理, 在构造正合同调列中有起到重要的作用. ...
Definition 1.9.1 A domain $R$ is a unique factorization domain (UFD) or factorial ring if every $r\in R$, neither $0$ or a unit, is a product of irreducibles; if $p_1\cdots p_…
Definition 1.8.1 If $a,b$ lie in a commutative ring $R$, then a greatest common divisor (gcd) of $a,b$ is a common divisor $d\in R$ which is divisible by every common divisor;…
Theorem 1.7.1 If $f(x)=a_0+a_1x+\cdots+a_nx^n\in \mathbb Z[x]\subset \mathbb Q[x]$, then every rational root of $f$ has the form $b/c$, where $b\mid a_0$ and $c\mid a_n$. In p…
Theorem 1.6.1 (Kronecker) If $k$ is a field and $f(x)\in k[x]$, there exists an extension field $K/k$ with $f$ a product of linear polynomials in $K[x]$. Proof. The proof is b…
Definition 1.5.1 An ideal $I$ in a commutative ring $R$ is called a maximal ideal if $I$ is a proper ideal for which there is no proper ideal $J$ with $I\subsetneq J$. Proposi…
Theorem 1.4.1 (Division Algorithm) If $K$ is a field and $f(x),g(x)\in K[x]$ with $f\ne 0$, then there are unique polynomials $q(x),r(X)\in K[x]$ with $$g(x)=q(x)f(x)+r(x), $$…