# Splitting field extension of degree $n!$

Suppose $$f \in K[X]$$ is a polynomial of degree n.

I had a small exercise were I had to prove that the degree of a field extension (by the splitting field of f which is $$\Sigma$$) $$[\Sigma : K]$$ divides $$n!$$. After convincing myself of this, I tried to find extensions, say of $$\mathbb{Q}$$ were we had in general the equality, i.e. the extension is of degree $$n!$$. What would be such an example?

## 2 Answers

In general the polynomials of the form $$x^n - px - p$$ for a prime integer $$p$$ give you such an example, but proving this in general isn't that "easy". If you are interested in a proof you can check the Wikipedia page on the more general topic of the Inverse Galois Problem. In fact your question is a special case of this problem, where you are asking whether each $$S_n$$ is realizable as a Galois group of a polynomial.

Here's a general theorem which fits your problem perfectly when $$n$$ is prime:

If $$f$$ is an irreducible polynomial of prime degree $$p$$ with rational coefficients and exactly two non-real roots, then the Galois group of $$f$$ is the full symmetric group $$S_p$$. [Wikipedia]