# How do these explanations for a Galois Group tie in at all?

In relation to my previous question (which I will get back), I am rather very confused with the various explanations people and textbooks have.

Firstly, I noticed there are $2$ different Galois Groups(or are they?),

$1.$ $Gal_{\mathbb{A}}(f)$ where $f$ is a polynomial over $\mathbb{A}$.

$2.$ $Gal_{\mathbb{K}}(L)$ for a field extension $L:K$ where $K,L$ are fields.

One is about polynomials over a field, another is talking about field extensions. Sure, an extended field might be a solution to a particular polynomial(i.e. splitting field) but so, are the $2$ Galois Groups the same? Different? No textbook states they're different but the trouble I have is, neither do they make any clear statement relating them. What on Earth are they?

So, what is a Galois group? Here are the $2$ "intuitive" explanations I have been given.