Polynomial Rings

Polynomials

Polynomial Ring

Let $R$ be a ring. The polynomial ring in one variable over $R$ is the set of sequences:$\{(a_n{)}_{n=0}^{\infty }\mid a_n\in R,\exists n_0\in \mathbb{N},a_n=0\text{ for all }n\ge n_0\}$Equivalently,$R[x]=\left\{{\sum }_{n=0}^{\infty }{a}_n{x}^n\mid a_n\in R,\exists n_0\in \mathbb{N},a_n=0\text{ for all }n\ge n_0\right\}$with addition $(a_n)+(b_n)=(a_n+b_n)$ and multiplication $(a_n)\cdot (b_n)=({\sum }_{k=0}^n{a}_k{b}_{n-k})$.

Proposition

$R[x]$ is a ring. The identity $1$ is the polynomial $(1+0x+0x^2+\dots )$.

Degree and Leading Term

The degree of a polynomial $f={\sum }_n{a}_n{x}^n\in R[x]$, denoted $\mathrm{deg}(f)$, is the largest $n$ with $a_n\ne 0$. The element zero does not have a degree.

Leading Term & Monic

The term $a_n{x}^n$ for $n=\mathrm{deg}(f)$ is called the leading term of $f$. The polynomial $f$ is called monic if the coefficient of the leading term is $1$.

Proposition

For all $f,g\in R[x]$, $\mathrm{deg}(fg)\le \mathrm{deg}(f)+\mathrm{deg}(g)$ with equality if and only if $R$ is a integral domain.

Proposition

$R[x{]}^{\times }=\{r+0x+0x^2+\cdots \mid r\in R\}\cong R^{\times }.$

Proposition

If $R$ is an integral domain, then $R[x]$ is an integral domain.

Division with Remainder

Theorem

Let $R$ be a ring and $f,g\in R[x]$ with $f$ monic. Then there are uniquely determined polynomials $q,r\in R[x]$, such that $g=f\cdot q+r$ and such that $\mathrm{deg}(r)<\mathrm{deg}(f)$ or $r=0$. The polynomial $q$ is called the quotient and the polynomial $r$ is called the remainder.

Def Substitution Principle I and Polynomial Function Let $R$ be a ring and $p=a_n{x}^n+\cdot \cdot \cdot +a_{1}x+a_0\in R[x]$. The substitution of $\alpha \in R$ in $p$ is the element$p(\alpha )=a_n{\alpha }^n+\cdot \cdot \cdot +a_1\alpha +a_0\in R$A function $f:R\to R$ is called a polynomial function if there exists $p\in R[x]$ such that $f(\alpha )=p(\alpha )$ for every $\alpha \in R$.

Corollary

Let $g\in R[x]$ and $\alpha \in R$. The remainder of division of $g(x)$ by $x-\alpha$ is $g(\alpha )$. It follows that $x-\alpha$ divides $g(x)$ if and only if $g(\alpha )=0$.

Proof We have unique $q,r\in R[x]$ such that $g(x)=(x-\alpha )q(x)+r(x)$ and $\mathrm{deg}(r)<\mathrm{deg}(f)=1$ or $r=0$. Hence $x-\alpha$ divides $g(x)$ if and only if $g(\alpha )=0$. $\square$

Corollary A polynomial in $F$ of degree $n$ over a field $F$ (more generally, a domain) has at most $n$ roots.

Multivariable Polynomials

For $x_1,\dots ,x_n$ and $i=(i_1,\dots ,i_n)\in {\mathbb{N}}^n$ we write $x^i=x_1^{i_1}\dots x_n^{i_n}$. The degree of $x^i$ is $i_1+\cdots +i_n$. The collection of all these is called the ring of polynomials in $n$ variables with coefficients in $R$ and denoted $R[x_1,\dots ,x_n]$.

Proposition

$\frac{R[x,y]}{(f(x),g(y))}\cong \frac{(R[x]/(f(x)))[y]}{g(y)}$

Prop Every ideal in the polynomial ring $F[x]$ over a field $F$ is principal. A non-zero ideal in $F[x]$ is generated by the unique monic polynomial of lowest degree that it contains.

Proof

Def Greatest Common Divisor Let $F$ be a field and $f,g\in F[x]$ be non-zero elements. The greatest common divisor $d\in F[x]$ of $f$ and $g$ is the unique monic polynomial that generates the ideal $(f,g)$. It has the properties:

1. $(d)=(f)+(g)$
2. $d\mid f,g$
3. If $e\mid f,g$ then $e\mid d$
4. There are polynomials $p,q$ such that $d=pf+qg$