Polynomial Rings

Throughout this chapter we shall assume that \(R\) is a commutative ring with identity. Any expression of the form \[\begin{aligned}\end{aligned}\], where \(a_i \in R\) and \(a_n \neq 0\), is called a polynomial over \(R\) with indeterminate \(x\). The elements \(a_0, a_1, \ldots, a_n\) are called the coefficients of \(f\). 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 \(a_n \neq 0\), we say that the degree of \(f\) is \(n\) and write \(\deg f(x) = n\). \(\deg f(x)\) degree of a polynomial If no such \(n\) existsthat is, if \(f=0\) is the zero polynomialthen the degree of \(f\) is defined to be \(-\infty\). We will denote the set of all polynomials with coefficients in a ring \(R\) by \(R[x]\). \(R[x]\) ring of polynomials over a ring \(R\) Two polynomials are equal exactly when their corresponding coefficients are equal; that is, if we let \[\begin{aligned}p(x) & = a_0 + a_1 x + \cdots + a_n x^n \\ q(x) & = b_0 + b_1 x + \cdots + b_m x^m\end{aligned}\], then \(p(x) = q(x)\) if and only if \(a_i = b_i\) for all \(i \geq 0\).

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 \[\begin{aligned}p(x) & = a_0 + a_1 x + \cdots + a_n x^n \\ q(x) & = b_0 + b_1 x + \cdots + b_m x^m\end{aligned}\]. Then the sum of \(p(x)\) and \(q(x)\) is \[\begin{aligned}\end{aligned}\], where \(c_i = a_i + b_i\) for each \(i\). We define the product of \(p(x)\) and \(q(x)\) to be \[\begin{aligned}\end{aligned}\], where \[\begin{aligned}\end{aligned}\] for each \(i\). Notice that in each case some of the coefficients may be zero.

Condensed: the full section is in Judson, Abstract Algebra: Theory and Applications.