If is a field and has degree , then there exists a splitting field of with .

**Proof:**

We use induction on .

Base case: If , or if splits over , then is a splitting field with .

Induction Hypothesis: Assume the statement is true for degree , where .

If and does not split over , let be an irreducible factor of with . Let be a root of , then

Write with of degree . By induction hypothesis, there exists a splitting field of over with .

That is, with and . Thus , so splits over .

This shows is a splitting field of over of dimension