Example 19 Factorize $x^{30}+1$ over the field of rational numbers.
Solve: $x^{30}+1=\left(x^{2}\right)^{15}+1=-\left[\left(-x^{2}\right)^{15}-1\right]$.
Notice that $x^{15}-1=F_{1}(x) \cdot F_{3}(x) \cdot F_{5}(x) \cdot F_{15}(x)$,
then
$$
\begin{array}{l}
-\left[\left(-x^{2}\right)^{15}-1\right]=-F_{1}\left(-x^{2}\right) \cdot F_{3}\left(-x^{2}\right) \cdot F_{5}\left(-x^{2}\right) \cdot F_{15}\left(-x^{2}\right), \\
\text { where }-F_{1}\left(-x^{2}\right)=-\left(-x^{2}-1\right)=x^{2}+1, \\
F_{3}\left(-x^{2}\right)=x^{4}-x^{2}+1, \\
F_{5}\left(-x^{2}\right)=x^{8}-x^{6}+x^{4}-x^{2}+1, \\
F_{15}\left(-x^{2}\right)=x^{16}+x^{14}-x^{10}-x^{8}-x^{6}+x^{2}+1 . \\
\text { Therefore, } x^{30}+1=\left(x^{2}+1\right)\left(x^{4}-x^{2}+1\right)\left(x^{8}-x^{6}+x^{4}-x^{2}+1\right) \cdot\left(x^{16}+x^{14}-x^{10}-\right. \\
\left.x^{8}-x^{6}+x^{2}+1\right) .
\end{array}
$$