Q1: What is the simplest example of two non-isomorphic fields $L$ and $K$ of characteristic $0$ such that $L(x)\simeq K(x)$ (here $x$ is an indeterminate)? 


Q2: Do we have a sufficient criterion for a general field $K$ of characteristic $0$ which guarantees that if $K(x\_1,\ldots,x\_n)\simeq L(x\_1,\ldots, x\_n)$ (here $L$ is a field and the
$x\_i$'s are indeterminates) then $K\simeq L$? 



non-isomorphic stably isomorphic fields
I don't think that there are any really easy examples. In the famous paper of Beauville, Colliot-Thélène, Sansuc and Swinnerton-Dyer "Variétés stablement rationnelles non rationnelles" they construct surfaces $S$ over $\mathbb Q$ that are not rational, but such that the products $S \times \mathbb P^3$ are rational. You get an example by taking $K$ to be a purely transcendental extension of the function field of $S$ of transcendence degree $d$, and a purely transcendental extension of $\mathbb Q$ of transcendence degree $d+2$, for some $d$ between $0$ and $3$ (I don't know the correct value of $d$).


