Polynomial Rings

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last updates: 14 February 2011

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 $$f\left(x\right)=r_0+r_{1}x+r_2{x}^2+\cdots$$ 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 $$r_i=s_i\text{for all}i=0,1,2,\dots$$
• The zero polynomial is the polynomial $0=0+0x+0x^2+\cdots .$
• 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 $f\left(x\right)=0+0x+0x^2+\dots$ 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 $$\begin{array}{c}f\left(x\right)+g\left(x\right)=(r_0+s_0)+(r_1+s_1)x+(r_2+s_2)x^2+\cdots ,\text{and}\\ f\left(x\right)g\left(x\right)=c_0+c_{1}x+c_2{x}^2+\cdots ,\text{where}{c}_k=\sum _{i+j=k}{r}_i{s}_j.\end{array}$$

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 $$\begin{array}{cccc}deg:& F\left[x\right]& \to & \mathbb{N}\\ & f\left(x\right)& \mapsto & deg\left(f\right(x\left)\right).\end{array}$$

Let $R,S$ be commutative rings and $\varphi :R\mapsto S$ be a ring homomorphism. Then the map $$\begin{array}{cccc}\psi :& R\left[x\right]& \to & S\left[x\right]\\ & r_0+r_{1}x+r_2{x}^2+\cdots & \mapsto & \phi \left(r_0\right)+\phi \left(r_1\right)x+\phi \left(r_2\right)x^2+\cdots \end{array}$$ is a ring homomorphism.

Adjoining elements to $R$, the rings

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 $$\begin{array}{cccc}{ev}_{\alpha }:& R\left[x\right]& \mapsto & R\\ & f\left(x\right)& \mapsto & f\left(\alpha \right),\end{array}$$ 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 . The ring $R$ adjoined $\alpha$ is the subring $R\left[\alpha \right]$ of $S$ given by $$R\left[\alpha \right]={ev}_{\alpha }\left(R\left[x\right]\right).$$

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 $$f\left(x\right)=cg\left(x\right).$$
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)

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