Roots of an irreducible polynomial over a finite field

[Scherie]

Let F=Z2 and let f(x) = X^3 +x+1 belong to F[x]. Suppose that a is a zero of f(x) in some extension of F.
Using the field created above F(a)
Show that a^2 and a^2+a are zeros of x^3+x+1?

 

[Deveno]

If $a$ is a root of $x^3 + x + 1 \in \Bbb Z_2[x]$, it follows that in $\Bbb Z_2(a)$, we have:

$a^3 = a + 1$ (recall that in any field of characteristic $2$, we have $b = -b$, for any element $b$).

It is straightforward to compute $f(a^2)$:

$f(a^2) = (a^2)^3 + (a^2) + 1 = (a^3)^2 + a^2 + 1 = (a+1)^2 + a^2 + 1$

$= a^2 + (a+a) + 1 + a^2 + 1 = a^2 + 0 + 1 + a^2 + 1 = (a^2 + a^2) + (1 + 1)$

$= a^2(1 + 1) + (1 + 1) = (a^2 + 1)(1 + 1) = (a^2 + 1)(0) = 0$.

This shows $a^2$ is likewise a root of $f$, since $f(a^2) = 0$.

Now can you do the same with $a^2 + a$?