Terminology for Galois groups of non-Galois extensions.

Question

I am having a confusion with terminology. If $L/K$ is not Galois, what is the meaning of "the Galois group of $L$ over $K$"? I have two guesses:

1) It is the field automorphisms of $L$ that fix $K$.

2) It is the Galois group of the Galois closure of $L$ over $K$.

Thanks!

Answer 1

As an example take the extension $F=\mathbb{Q}(\theta)$ where $\theta=\sqrt[3]{2}$, it is a field extension of degree 3 with basis $\{1, \theta, \theta^2\}$. But it is not the splitting field of $x^3-2$ (see cubic example), it does not contain all the roots of $x^3-2$. Every element of the Galois group associated with this polynomial would map $F$ "outside" of F so can't be considered an automorphism of $F$. If $p$ is a polynomial having a square integer as discriminant then $F$ would be the splitting field of $p$, and the Galois group is the cyclic group of order $3$.

Answer 2

Indeed, (2) seems to be standard convention. Wikipedia defines it that way, and I have run across this and become confused before myself.