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…

Definition 1.1.1 If $R$ is a commutative ring, then a formal power series over $R$ is a sequence of elements $s_i\in R$ for all $i\ge 0$, called the coefficients of $\sigma$: …

Let's begin with commutative rings. Commutative rings are algebra structures which have good properties and close to some sets we have learned well, such as $\mathbb {R}$ and …

代数学基本定理即在域 $\mathbb C$ 中，$n$ 次多项式方程有且仅有 $n$ 个根。 ...