## You are here

Homepolynomial ring

## Primary tabs

# polynomial ring

# 1 Polynomial rings in one variable

Let $R$ be a ring. The *polynomial ring* over $R$ in one variable $X$ is the set $R[X]$ of all sequences in $R$ with only finitely many nonzero terms. If $(a_{0},a_{1},a_{2},a_{3},\dots)$ is an element in $R[X]$, with $a_{n}=0$ for all $n>N$, then we usually write this element as

$\sum_{{n=0}}^{N}a_{n}X^{n}=a_{0}+a_{1}X+a_{2}X^{2}+a_{3}X^{3}+\cdots+a_{N}X^{N}.$ |

Elements of $R[X]$ are called *polynomials* in the indeterminate $X$ with coefficients in $R$. The ring elements $a_{0},\ldots,a_{N}$ are called *coefficients* of the polynomial, and the *degree* of a polynomial is the largest natural number $N$ for which $a_{N}\neq 0$, if such an $N$ exists. When a polynomial has all of its coefficients equal to $0$, its degree is usually considered to be undefined, although some people adopt the convention that its degree is $-\infty$.

A *monomial* is a polynomial with exactly one nonzero coefficient. Similarly, a *binomial* is a polynomial with exactly two nonzero coefficients, and a *trinomial* is a polynomial with exactly three nonzero coefficients.

Addition and multiplication of polynomials is defined by

$\displaystyle\sum_{{n=0}}^{N}a_{n}X^{n}+\sum_{{n=0}}^{N}b_{n}X^{n}$ | $\displaystyle=$ | $\displaystyle\sum_{{n=0}}^{N}(a_{n}+b_{n})X^{n}$ | (1) | ||

$\displaystyle\sum_{{n=0}}^{N}a_{n}X^{n}\cdot\sum_{{n=0}}^{N}b_{n}X^{n}$ | $\displaystyle=$ | $\displaystyle\sum_{{n=0}}^{{2N}}\left(\sum_{{k=0}}^{n}a_{k}b_{{n-k}}\right)X^{n}$ | (2) |

$R[X]$ is a $\mathbb{Z}$–graded ring under these operations, with the monomials of degree exactly $n$ comprising the $n^{\mathrm{th}}$ graded component of $R[X]$. The zero element of $R[X]$ is the polynomial whose coefficients are all $0$, and when $R$ has a multiplicative identity $1$, the polynomial whose coefficients are all $0$ except for $a_{0}=1$ is a multiplicative identity for the polynomial ring $R[X]$.

# 2 Polynomial rings in finitely many variables

The *polynomial ring* over $R$ in two variables $X,Y$ is defined to be $R[X,Y]:=R[X][Y]\cong R[Y][X]$. Elements of $R[X,Y]$ are called *polynomials* in the indeterminates $X$ and $Y$ with coefficients in $R$. A *monomial* in $R[X,Y]$ is a polynomial which is simultaneously a monomial in both $X$ and $Y$, when considered as a polynomial in $X$ with coefficients in $R[Y]$ (or as a polynomial in $Y$ with coefficients in $R[X]$). The *degree* of a monomial in $R[X,Y]$ is the sum of its individual degrees in the respective indeterminates $X$ and $Y$ (in $R[Y][X]$ and $R[X][Y]$), and the degree of a polynomial in $R[X,Y]$ is the supremum of the degrees of its monomial summands, if it has any.

In three variables, we have $R[X,Y,Z]:=R[X,Y][Z]=R[X][Y][Z]\cong R[X][Z][Y]\cong\cdots$, and in any finite number of variables, we have inductively $R[X_{1},X_{2},\dots,X_{n}]:=R[X_{1},\dots,X_{{n-1}}][X_{n}]=R[X_{1}][X_{2}]% \cdots[X_{n}]$, with monomials and degrees defined in analogy to the two variable case. In any number of variables, a polynomial ring is a graded ring with $n^{\mathrm{th}}$ graded component equal to the $R$-module generated by the monomials of degree $n$.

# 3 Polynomial rings in arbitrarily many variables

For any nonempty set $M$, let $E(M)$ denote the set of all finite subsets of $M$. For each element $A=\{a_{1},\ldots,a_{n}\}$ of $E(M)$, set $R[A]:=R[a_{1},\ldots,a_{n}]$. Any two elements $A,B\in E(M)$ satisfying $A\subset B$ give rise to the relationship $R[A]\subset R[B]$ if we consider $R[A]$ to be embedded in $R[B]$ in the obvious way. The union of the rings $\{R[A]:A\in E(M)\}$ (or, more formally, the categorical direct limit of the direct system of rings $\{R[A]:A\in E(M)\}$) is defined to be the ring $R[M]$.

## Mathematics Subject Classification

11C08*no label found*12E05

*no label found*13P05

*no label found*17B66

*no label found*16W10

*no label found*70G65

*no label found*17B45

*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff
- Corrections

## Attached Articles

a polynomial of degree $n$ over a field has at most $n$ roots by alozano

polynomial ring over integral domain by pahio

algebraic equation by PrimeFan

square root of polynomial by pahio

derivative of polynomial by pahio

how to multiply polynomials by Algeboy

explicit generators of a quotient polynomial ring associated to a given polynomial by joking

polynomial analogon for Fermat's last theorem by pahio

explicit definition of polynomial rings in arbitrarly many variables by joking