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 $$g(x)=q(x)f(x)+r(x), $$…
Definition 1.3.1 Let $I$ be an ideal in a commutative ring $R$. If $a\in R$, then the coset $a+I$ is the subset $$a+I=\{a+i;i\in I\}. $$ The coset $a+I$ is often called $a\bmo…
Homomorphisms allow us to compare rings. Definition 1.2.1 If $A$ and $R$ are (not necessarily commutative) rings, a (ring) homomorphism is a function $\varphi :A\to R$ such th…