How to decompose the polynomial $x^{21} + 1$ into a product of irreducible factors over $\mathbb Z/2\mathbb Z$? This is a problem from a past abstract algebra exam, the degree $21$ was too high for me to solve it. No Wolfram Mathematica please!

$$x^{21} + 1 = x^{21} - 1 = (x^7)^3 - 1 = (x^7 - 1)(x^{14} + x^7 + 1) = (x - 1)(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)(x^{14} + x^7 + 1) = (x -1)(x^6 + x^5 + x^4 + x^3 + x^2 + x + 1)(x^2 + x + 1)(x^{12} - x^{11} + x^9 - x^8 + x^6 - x^4 + x^3 - x +1)$$
Now: The $2$nd one can be shown to be irreducible in $\mathbb Z[x]$; I don't know about $\frac{\mathbb Z}{2\mathbb Z}[x]$. The $3$rd one can be shown to be irreducible in $\frac{\mathbb Z}{2\mathbb Z}[x]$. Good luck with the last one.
