Problem 10

Let $F$ be a field and $f(x)$ is a non-constant polynomial in $F[x]$ with degree $n$. Let $E$ be the splitting field of $f(x)$ over $F$.

1. By considering the binomial coefficient ${a+b\choose a}$, explain why $a!b!$ divides $(a+b)!$ for all non-negative integers $a$ and $b$.

2. Prove that $[E:F]$ divides $n!$.(Hint: use strong induction on $n$; break into two cases, where $f(x)$ is irreducible or $f(x)$ is reducible.)