Suppose we have two field extensions $K/k$ and $L/k$. I am able to show that if these field extensions are isomorphic, then their corresponding Galois groups Aut$(K/k)$ and Aut$(L/k)$ are also isomorphic.
However, does the converse hold true? Every example I've been able to come up with has suggested that the converse is true, but I do not believe it to be so.
The converse is false: let $k=\mathbb{Q}$, $K=\mathbb{Q}(\sqrt{2})$, and $L=\mathbb{Q}(\sqrt{3})$. Then $\mathrm{Gal}(K/k)\simeq \mathrm{Gal}(L/k)\simeq \mathbb{Z}/2\mathbb{Z}$, but $K$ and $L$ are not isomorphic, since $K$ contains a square root of $2$ but $L$ doesn't.