This is homework exercise: $$P=t^{1024} + t + 1 , R = \mathbf{F}_{2}[t] \Rightarrow P \ \text{reducible in R}$$
I wanted to show this analogous to how a book shows it (book shows it with other numbers and field): $a^{1024}= a+1$ has solutions in $R/PR$, then calculate $a^{2^{20}}, a^{2^{1024}}$ in $R/PR$ and by using gcd conclude that P is reducible over R.
My problem is that the numbers are so big that I can't split up and show that there are solutions or calculate $a^{2^{20}}, a^{2^{1024}}$ in $R/PR$. But there must be an easy way since it is homework. So how to show that $a^{1024} = a+1$ is solvable in $R/PR$ and how to compute $a^{2^{20}}, a^{2^{1024}}$ in $R/PR$? Thanks for all input.