Minimal Polynomial over a finite field

Question

Suppose $\alpha$ satisfies the equation $\alpha^6 + \alpha + 1 = 0$ over $\mathbb{Z}_2$. Find the minimal polynomial of $\alpha^{21}$ over $\mathbb{Z}_2$.

I tried this:

Since $\alpha^6 + \alpha + 1 = 0$, therefore $\alpha^{21} + \alpha^5 + \alpha^4 + \alpha^3 + \alpha + 1 = 0$, but from here i couldn't get much more to find the minimal polynomial of $\alpha^{21}$ over $\mathbb{Z}_2$. What i have to do then?

Answer 1

Hints:

1. What is the cardinality of $\mathbb{F}_2(\alpha)$? What is the cardinality of the group of units $\mathbb{F}_2(\alpha)^\times$?

2. Thus $\alpha^{63} = 1$ by ________ Theorem. So $$ 0 = \alpha^{63} - 1 = (\alpha^{21})^3 - 1 = \cdots ? $$

3. Your calculations show that $\alpha^{21} \neq 1$. Why?

4. Can you prove that $X^2 + X + 1$ is the minimal polynomial of $\alpha^{21}$ over $\mathbb{F}_2$? Why is it irreducible?