What is the Degree of a Splitting Field for a Polynomial over a Field?

[mufq15]

Homework Statement

Let f in K[x] be a polynomial over a field K. Define the notion of a splitting field
L of f over K. Show that if deg f = d, then f has a splitting field over K of degree
dividing d!

The Attempt at a Solution

If f is reducible, then this seems true by induction. I'm not sure about the case where f is irreducible over K though.

Thanks for your help.

 

[mufq15]

For anyone keeping score at home, I think the irreducible case is also by induction. Consider an intermediate field made by adjoining a root r of f to K, and then consider L as a splitting field over K(r) of g(x) (where f(x) = (x - r)g(x).)