Leading Coefficient of Product of Polynomials over Integral Domain

Theorem

Let $R$ be an integral domain.

Let $f, g \in R \sqbrk x$ be polynomials.

Let $c$ and $d$ be their leading coefficients.

Then $f g$ has leading coefficient $c d$.

Proof

Let $p = \deg f$ and $q = \deg g$ be their degrees.

By Degree of Product of Polynomials over Integral Domain, $\map \deg {f g} = p + q$.

For a natural number $n \ge 0$, let:

$a_n$ be the coefficient of the monomial $x^n$ in $f$.

$b_n$ be the coefficient of the monomial $x^n$ in $g$.

By Coefficients of Product of Two Polynomials, the coefficient of $x^{p + q}$ in $f g$ is the sum:

$\ds \sum_{k \mathop = p}^p a_k b_{p + q - k}$

By Indexed Summation over Interval of Length One:

$\ds \sum_{k \mathop = p}^p a_k b_{p + q - k} = a_p b_q = c d$

$\blacksquare$