Polynomial Rings

## Polynomial rings

Definition.

• Let $R$ be a commutative ring and for each $i=1,2,3,\dots$ let ${x}^{i}$ be a formal symbol. A polynomial with coefficients in in $R$ is an expression of the form $fx= r0+r1x +r2x2+⋯$ such that
1. ${r}_{i}\in R$ for $i=0,1,2,\dots ,$
2. There exists a positive integer $N$ such that ${r}_{i}=0$ for all $i>N$.
• Let $R$ be a commutative ring. Polynomials $f\left(x\right)={r}_{0}+{r}_{1}x+{r}_{2}{x}^{2}+\cdots$ and $g\left(x\right)={s}_{0}+{s}_{1}x+{s}_{2}{x}^{2}+\cdots$ with coefficients in $R$ are equal if $ri=si for alli=0,1,2,…$
• The zero polynomial is the polynomial $0=0+0x+0{x}^{2}+\cdots .$
• The degree, $deg\left(f\left(x\right)\right)$, of a polynomial $f\left(x\right)={r}_{0}+{r}_{1}x+{r}_{2}{x}^{2}+\cdots$ with coefficients in $R$ is the smallset nonnegative integer $N$ such that ${r}_{N}\ne 0$ and ${r}_{k}=0$ for all $k>N$. If $f\left(x\right)=0+0x+0{x}^{2}+\dots$ then we define $deg\left(f\left(x\right)\right)=0$.
• Let $R$ be a commutative ring. The ring of polynomials withe coefficients in $R$ is the set $R\left[x\right]$ of polynomials with coefficients in $R$ with the operations of addition and multiplication as defined as follows:

If $f\left(x\right),g\left(x\right)\in R\left[x\right]$ where $f\left(x\right)={r}_{0}+{r}_{1}x+{r}_{2}{x}^{2}+\cdots$ and $g\left(x\right)={s}_{0}+{s}_{1}x+{s}_{2}{x}^{2}+\cdots ,$ then $fx+gx= (r0+s0)+ (r1+s1)x+ (r2+s2)x2 +⋯, and f(x)g(x)= c0+c1x +c2x2+⋯, where ck= ∑ i+j=k risj.$

Let $R$ be a commutative ring. Then $R\left[x\right]$ is a commutative ring.

Let $R$ be an integral domain. Then $R\left[x\right]$ is an integral domain.

The following theorem is a deep theorem which will be proved at the end of this section.

Let $R$ be a unique factorization domain. Then $R\left[x\right]$ is a unique factorization domain.

The following is an important theorem which shows that if $F$ is a field then $F\left[x\right]$ is a Euclidean domain, and therefore, by Theorem 1.3, that $F\left[x\right]$ is a principal ideal domain.

Let $F$ be a field. The ring $F\left[x\right]$ is a Euclidean domain with size function given by $deg: F[x] → ℕ f(x) ↦ deg(f(x)).$

Let $R,S$ be commutative rings and $\varphi :R↦S$ be a ring homomorphism. Then the map $ψ: R[x] → S[x] r0+r1x+r2x2 +⋯ ↦ φ(r0)+φ(r1)x+φ(r2)x2 +⋯$ is a ring homomorphism.

Adjoining elements to $R$, the rings $R\left[\alpha \right]$

Definition.

• Let $R$ be a commutative ring and let $\alpha \in R$. The evaluation map ${ev}_{\alpha }:R\left[x\right]\to R$ is the map given by $evα: R[x] ↦ R f(x) ↦ f(α),$ where, if $f\left(x\right)={r}_{0}+{r}_{1}x+{r}_{2}{x}^{2}+\dots$ then $f\left(\alpha \right)={r}_{0}+{r}_{1}\alpha +{r}_{2}{\alpha }^{2}+\dots .$

Let $R$ be a commutative ring and let $\alpha \in R$. Then the evaluation homomorphism ${ev}_{\alpha }:R\left[x\right]\to R$ is a ring homomorphism.

Definition

• Let $S$ be a commutative ring. Let $R\subseteq S$ be a subring and let $\alpha \in S$. Let ${ev}_{\alpha }:S\left[x\right]\to S$ be the evaluation homomorphism given by evaluating at $\alpha$. The ring $R$ adjoined $\alpha$ is the subring $R\left[\alpha \right]$ of $S$ given by $R[α]= evα(R[x]).$

HW: Prove that $R\left[\alpha \right]={ev}_{\alpha }\left(R\left[x\right]\right).$ is a subring of $S$.

HW: Let $S$ be a commutative ring. Let $R\subseteq S$ be a subring and let $\alpha \in S$ show that the ring $R\left[\alpha \right]$ is the subring of $S$ consisting of all elements of the form ${r}_{0}+{r}_{1}x+{r}_{2}{x}^{2}+\cdots +{r}_{d}{x}^{d},$ where ${r}_{i}\in R$ and $d$ is a nonnegative integer.

Proof of Theorem 1.3

Definition. Let $R$ be a unique factorization domain. A polynomial $f\left(x\right)={c}_{0}+{c}_{1}x+\cdots +{c}_{k}{x}^{k}\in R\left[x\right]$ is primitive if there does not exist any $p\in R$ such that $p$ divides all of the coefficients ${c}_{0},{c}_{1},\dots {c}_{k},$ of $f\left(x\right)$.

(Gauss' Lemma) Let $R$ be a unique factorization domain. Let $f\left(x\right),g\left(x\right)\in R\left[x\right]$ be primitive polynomials. Then $f\left(x\right)g\left(x\right)$ is a primitive polynomial.

Let $R$ be a unique factorization domain. Let $F$ be the field of fractions of $R$ and let $f\left(x\right)\in F\left[x\right]$. Then

1. There exists an element $c\in F$ and a primitive polynomial $g\left(x\right)\in R\left[x\right]$ such that $f(x)=cg(x).$
2. The factors $c$ and $g\left(x\right)$ are unique up to multiplication by a unit.
3. $f\left(x\right)$ is irreducible in $F\left[x\right]$ if and only if $g\left(x\right)$ is irreducible in $R\left[x\right]$.

Let $R$ be a unique factorization domain. Then $R\left[x\right]$ is a unique factorization domain.

## References

[CM] H. S. M. Coxeter and W. O. J. Moser, Generators and relations for discrete groups, Fourth edition. Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], 14. Springer-Verlag, Berlin-New York, 1980. MR0562913 (81a:20001)

[GW1] F. Goodman and H. Wenzl, The Temperly-Lieb algebra at roots of unity, Pacific J. Math. 161 (1993), 307-334. MR1242201 (95c:16020)