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

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?

• We need more information. Because the order of $S_n$ is larger than $n$, a straight forward interpretation doesn't feel right. It could mean that the Galois group is associated to the normal closure of $K/\Bbb{Q}$. But in the latter case it is more (?) normal to talk about the Galois group of the minimal polynomial of a generating element $\alpha$ such that $K=\Bbb{Q}(\alpha)$. We really need more surrounding context. I'm sure there are clues. – Jyrki Lahtonen Jun 12 '18 at 10:47

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 $\mathbb{Q}$.
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.