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 $p_1\cdots p_… Theorem 1.7.1 If$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 $\sigma$: …