Rewrite $a^{20}-1$ as $\left(a^{10}\right)^{2}-1^{2}$. The difference of squares can be factored using the rule: $p^{2}-q^{2}=\left(p-q\right)\left(p+q\right)$.
$$\left(a^{10}-1\right)\left(a^{10}+1\right)$$
Consider $a^{10}-1$. Rewrite $a^{10}-1$ as $\left(a^{5}\right)^{2}-1^{2}$. The difference of squares can be factored using the rule: $p^{2}-q^{2}=\left(p-q\right)\left(p+q\right)$.
$$\left(a^{5}-1\right)\left(a^{5}+1\right)$$
Consider $a^{5}-1$. By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where $p$ divides the constant term $-1$ and $q$ divides the leading coefficient $1$. One such root is $1$. Factor the polynomial by dividing it by $a-1$.
Consider $a^{5}+1$. By Rational Root Theorem, all rational roots of a polynomial are in the form $\frac{p}{q}$, where $p$ divides the constant term $1$ and $q$ divides the leading coefficient $1$. One such root is $-1$. Factor the polynomial by dividing it by $a+1$.
Consider $a^{10}+1$. Find one factor of the form $a^{k}+m$, where $a^{k}$ divides the monomial with the highest power $a^{10}$ and $m$ divides the constant factor $1$. One such factor is $a^{2}+1$. Factor the polynomial by dividing it by this factor.
Rewrite the complete factored expression. The following polynomials are not factored since they do not have any rational roots: $a^{4}-a^{3}+a^{2}-a+1,a^{4}+a^{3}+a^{2}+a+1,a^{8}-a^{6}+a^{4}-a^{2}+1,a^{2}+1$.
