The splitting field of an irreducible polynomial of degree n over F is a degree п.

Theorem. Let F be a field and f e F[x] a polynomial of degree n. Then there is a finite-dimensional extension of F in which ƒ factors into linear factors f(x) = (x – a1) ... (x – an). Proof. Apply the last theorem repeatedly to get getting extensions of extensions and factor out a linear factor each time until the degree is reduced to 1.