What is the meant by Galois group associated the number field?

Question

In the article I am reading it says: Let $K$ be a number field where $[K : \mathbb{Q}] = n$. Suppose the associated Galois group is $S_n$ (the symmetric group).

I was wondering what is meant by ``the associated Galois group'' here?

Answer 1

The associated Galois group is the Galois group of the normal closure of $K$. So in your context, $K$ has degree $n$ over $\mathbb{Q}$ and is not Galois, and its Galois closure has Galois group $S_n$ (as big as it can be) over $\mathbb{Q}$.

Answer 2

I think the "associated Galois group" must mean the Galois group $Gal(K/ \mathbb{Q})$, which is the group of automorphisms of $K$ that fix $\mathbb{Q}$ pointwise.