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