Generic polynomial (awaiting review)

Let $K$ be a field and $G$ a finite group, $t_1, \ldots, t_n$ and $x$ indeterminants. A polynomial $f(t_1,\ldots,t_n,x)\in K(t_1,\ldots,t_n)[x]$ is generic for G if

1. the splitting field $L$ of $f$ is Galois and regular over $K(t_1,\ldots,t_n)$
2. $\Gal(L/K(t_1,\ldots,t_n))\cong G$
3. every Galois extension $E/K$ with $\Gal(E/K)\cong G$ is the splitting field of $f(a_1,\ldots,a_n,x)$ for some $(a_1,\ldots,a_n)\in K^n$
4. condition 3 holds for every extension field of $K$.