# Determine degree min. polynomial

I need a check on the following question

Let $$\alpha$$ a primitive element of $$\mathbb{F}_{2^n}$$. Determine the degree of the minimal polynomial over $$\mathbb{F}_2$$. What can you say about the splitting field?

From theory I know that the degree $$d$$ of the min, poly is the minimum integer such that $$\alpha^{2^d}=1$$. Now, $$\alpha$$ is primitive, so this means that $$\alpha^{2^n} = 1$$, hence the the min. poly has degree $$2^n$$.

Then, since the min. poly is irreducible, by definition, I have that the splitting field of $$h$$ over $$\mathbb{F}_2$$ is given by $$\mathbb{F}_{2^{2^n}}$$

Is everything okay?

• No: you know $\alpha^{2^d-1}=1$ by definition of $d$, and $\alpha^{2^n-1}=1$. So $d | n$. In particular $d$ cannot equal $2^n$. – Mindlack Jun 2 at 14:46
• Yes, you're right @Mindlack So, what could be the degree? I can't stil find an answer – lukk Jun 2 at 15:05
• Of course, $d=n$... And the splitting field is $\mathbb{F}_{2^n}$. Right @Mindlack ? – lukk Jun 2 at 15:06
• Yes, that's it. – Mindlack Jun 2 at 16:23