Exercise 11.2ΒΆ

Let \(F\) be a field and let \(a, b \in F[x]\), with both nonzero. We want to show that \(\deg(a) \leq \deg(ab)\).

Consider two nonzero polynomials \(a, b\) and think about their product \(ab\). We already know that the leading term of \(ab\) comes from the product of the leading terms of \(a\) and \(b\), whose powers of \(x\) will be \(\deg(a)\) and \(\deg(b)\) respectively. So the power of \(x\) in the leading term of \(ab\) is \(\deg(a) + \deg(b)\), i.e. \(\deg(ab) = \deg(a) + \deg(b)\).

So our original inequality is equivalent to \(\deg(a) \leq \deg(a) + \deg(b)\) or equivalently, \(0 \leq \deg(b)\). But we know this to be true already: the degree of a nonzero polynomial is always nonnegative! So we are done.