I need to prove, specifically using modules over $\Bbb{Z}$, that the multiplicative group of a finite field is cyclic. This is what I've done already:
The multiplicative group of a finite field is in particular an abelian group. Hence, a $\Bbb{Z}$-module. Because $\Bbb{Z}$ is a PID, we can write the multiplicative group as $$ \Bbb{F}_{p^n}^\times \cong \Bbb{Z}/(c_1) \oplus \ldots \oplus \Bbb{Z}/(c_k) $$ with $c_1 \vert c_2 \vert \ldots \vert c_k$. Equivalently, $$ \Bbb{F}_{p^n}^\times \cong \Bbb{Z}/(p_1^{a_1}) \oplus \ldots \oplus \Bbb{Z}/(p_m^{a_m}) $$ where the $p_i$ are prime elements and $a_i$ are positive integers. (The $p_i$ need not be distinct).
I don't know how to go from here. I know eventually we must have that the group is isomorphic to $\Bbb{Z}/(p^n-1)$ but I don't know how that's true. Do I use the Chinese Remainder Theorem? We aren't working with coprime numbers here?
Thanks in advance.