Definition:Multiplication of Polynomial Forms

Definition

Let $\displaystyle f = \sum_{k \in Z} a_k \mathbf X^k$, $\displaystyle g = \sum_{k \in Z} b_k \mathbf X^k$ be polynomials in the indeterminates $\left\{{X_j: j \in J}\right\}$ over $R$.

We define the product:

$\displaystyle f \circ g := \sum_{k \in Z} c_k \mathbf X^k$

where:

$\displaystyle c_k = \sum_{\substack{p + q = k \\ p, q \in Z}} a_p b_q$

It follows from Polynomials Closed under Ring Product that $f \circ g$ is a polynomial.

Sources

• C.R.J. Clapham: Introduction to Abstract Algebra (1969)... (previous)... (next): $\S 6.25$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 64$