In a proof I was reading on how the Frobenius automorphism generates the Galois group of $GF(p^n)$ over $GF(p)$, it says that :
$\alpha \in GF(p^n)$, the element which we adjoin to $GF(p)$ to get $GF(p^n)$ has minimum polynomial
suppose the minimal polynomial of α has degree t ; then it has the following form:
$$m(x)=(x-\alpha)(x-\alpha^{p})(x-\alpha^{p^2})……(x-\alpha^{p^{t-1}})$$
How is this known ? I looked back at my rings and field course and my linear algebra course but it wasn't proven there either.
Many thanks for any help !
Remark: I'll prove the more general claim that if $\alpha \in GF(P^n)$ then it's minimal polynomial can be factored as $(x-\alpha)(x-\alpha^p)\cdots(x-\alpha^{p^{t-1}})$, where $t$ is the degree of the minimal polynomial of $\alpha$ over $GF(P)$. If you take $\alpha$ to be the element we adjoin to $GF(P)$ in order to obtain $GF(P^n)$ we have that $t=n$ and everything else is the same.
Use the fact that the $GF(P)$ has characteristic $p$ and so we have that $a^p + b^p = (a+b)^p$ for any $a,b \in GF(P)$. In particular this gives you $a^p = a$ for $a \in GF(P)$.
Now let $x^t + a_1x^{t-1} + \cdots + a_t = 0$ be the minimal polynomial of $\alpha$ over $GF(P)$. Then note that $a^p$ is a root too. Indeed using what I mentioned above we have:
$$(\alpha^p)^t + a_1(\alpha^p)^{t-1} + \cdots + a_t = (\alpha^t)^p + (a_1\alpha^{t-1})^p + \cdots + a_t^p = (\alpha^t + a_1\alpha^{t-1} + \cdots + a_t)^p = 0$$
In a similar manner you have that $(\alpha^p)^p = \alpha^{p^2}$ is a root, too. As well as $\alpha^{p^k}$ for any integer $k$.
Finally, you need to prove that $\alpha^{p^i} \not = \alpha^{p^j}$ for $0 \le i< j \le t-1$. Otherwise we have $\alpha^{p^j - p^i} - 1 = 0$. This means that $\alpha$ has order less than $p^{t-1} - 1$. However adjoining $\alpha$ to $GF(P)$ yields $GF(P^t)$ and $\alpha$ is the generator of the group of units of $GF(P^t)$, which has order $p^t - 1$. Thus $\alpha$ has order $p^t - 1$, which contradicts the claim above.
Hence $\{\alpha^{p^k} \mid 0 \le k \le t-1 \}$ are the distinct roots of the minimal polynomial of $\alpha$ over $GF(P)$.