Abstract Algebra: Theory and Applications

Throughout this chapter we shall assume that $R$ is a commutative ring with identity. Any expression of the form

where $a_i\in R$ and $a_n\ne 0\text{,}$ is called a polynomial over $R$ with indeterminate $x\text{.}$ The elements $a_0,a_1,\dots ,a_n$ are called the coefficients of $f\text{.}$ The coefficient $a_n$ is called the leading coefficient. A polynomial is called monic if the leading coefficient is 1. If $n$ is the largest nonnegative number for which $a_n\ne 0\text{,}$ we say that the degree of $f$ is $n$ and write $\mathrm{deg}f(x)=n\text{.}$ If no such $n$ exists—that is, if $f=0$ is the zero polynomial—then the degree of $f$ is defined to be $-\mathrm{\infty }\text{.}$ We will denote the set of all polynomials with coefficients in a ring $R$ by $R[x]\text{.}$ Two polynomials are equal exactly when their corresponding coefficients are equal; that is, if we let

then $p(x)=q(x)$ if and only if $a_i=b_i$ for all $i\ge 0\text{.}$

To show that the set of all polynomials forms a ring, we must first define addition and multiplication. We define the sum of two polynomials as follows. Let

Then the sum of $p(x)$ and $q(x)$ is

where $c_i=a_i+b_i$ for each $i\text{.}$ We define the product of $p(x)$ and $q(x)$ to be

where

for each $i\text{.}$ Notice that in each case some of the coefficients may be zero.