Let $K$ be a field and let $f(t)$ be a monic polynomial with coefficients in $K$. What is meant by a splitting field $L$ for $f(t)$ over $K$ ? Show that such a splitting field exists and is unique up to isomorphism.

Now suppose that $K$ is a finite field. Prove that $L$ is a Galois extension of $K$ with cyclic Galois group. Prove also that the degree of $L$ over $K$ is equal to the least common multiple of the degrees of the irreducible factors of $f(t)$ over $K$.

Now suppose $K$ is the field with two elements, and let

$$\mathcal{P}_{n}=\{f(t) \in K[t] \mid f \text { has degree } n \text { and is irreducible over } K\}$$

How many elements does the set $\mathcal{P}_{9}$ have?