Galois extension definition. Let $L,K$ be fields with $L/K$ a field extension. We say $L/K$ is a Galois extension if $L/K$ is normal and separable. 
I don't fully understand this definition, is it saying that 
1) $L$ has to be the splitting field for some polynomial in $K[x]$ and that polynomial must not have any repeated roots, or is it saying that 
2) $L$ has to be the splitting field for all polynomials in $K[x]$ and all polynomials must not have repeated roots?

the extension $L/K$ is galoisien is equivalently to each $x\in L$, 
$x$ is algebraic over $K$ and the minimal polynomial of $x$ has
simple roots and are all in $L$. so by primitive element theorem
we get: the extension $L/K$  finite and  galoisienne is equivalent
to your proposal 1) but not to 2).
but in the infinity case; $L/K$ galoisienne  is not equivalent
either to 1) or 2)
