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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?

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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.

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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]

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