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!