Let $K$ be a field, and $G$ a finite subgroup of $K^{*}$. Show that $G$ is cyclic.

Define the cyclotomic polynomials $\Phi_{m}$, and show from your definition that

$$X^{m}-1=\prod_{d \mid m} \Phi_{d}(X)$$

Deduce that $\Phi_{m}$ is a polynomial with integer coefficients.

Let $p$ be a prime with $(m, p)=1$. Let $\Phi_{m} \equiv f_{1} \ldots f_{r} \quad(\bmod p)$, where $f_{i} \in \mathbb{F}_{p}[X]$ are irreducible. Show that for each $i$ the degree of $f_{i}$ is equal to the order of $p$ in the group $(\mathbb{Z} / m \mathbb{Z})^{*}$.

Use this to write down an irreducible polynomial of degree 10 over $\mathbb{F}_{2}$.