## Tag Archives: polynomial

### Dimensionality of a splitting field of a polynomial of degree n

$\mathbf{Problem:}$ Show that the dimensionality of a splitting field $E/F$ of $f(x)$ of degree $n$ is at most $n!$.
$\mathbf{Proof:}$ Assume $n\geq 1$(otherwise $E=F$). There exists an extension $K$ of $F$ containing a root $\alpha$ of $f(x)$.Consider the extension $F(\alpha)\subset K$ of $F$, then the minimal polynomial of $\alpha$ in $F(\alpha)$ must divide $f(x)$, which has degree $n$, then $[F(\alpha):F]\leq n$. Suppose $f(x)=(x-\alpha)g(x)$, with $g(x)\in F(\alpha)[x]$, $deg(g)=n-1$. By induction, $[E:F(\alpha)]\leq(n-1)!$, so

$\displaystyle [E:F]=[E:F(\alpha)][F(\alpha):F]\leq n!$

$\text{Q.E.D.}\hfill\square$