Do Isomorphic Galois Groups Guarantee Identical Field Extensions?

[mathbalarka]

Now this is totally a mad problem, but I am going to post it here anyways

It is pretty well-known that there are distinct fields over $\Bbb Q$ with isomorphic Galois groups. Take, for example, $K_1 = \Bbb Q(\sqrt{2})$ and $K_2 = \Bbb Q(\sqrt{5})$. Both have Galois groups of order $2$ as easily can be seen and there is only one group of order $2$ which is $\Bbb Z_2$.

Now take some algebraically closed field $k$. I suspect that if two field extensions $K/k(t)$ and $K'/k(t)$ are Galois over $k(t)$ with isomorphic galois groups, it's necessary that $K = K'$. Is this true?

In particular, take up $\Bbb C(z)$ and let $$\mathcal{G}(K/\Bbb C(z)) \cong \mathcal{G}(K'/\Bbb C(z))$$ Is it possible to prove/disprove that $K = K'$?

Here is a possible way to do it using non-classical galois theory : $\Bbb C(z)$ forms the function field of meromorphic functions of single variable over $\Bbb P^1$, thus any Galois group of some Galois extension is realized as monodromy of coverings $X \to \Bbb P^1$ and $X_0 \to \Bbb P^1$. These are both galois coverings, so $\Bbb P^1 \cong X/G$ and $\Bbb P^1 \cong X_0/G$ G being the monodromy of both (the galois groups are isomorphic). So $X/G \cong X_0/G$. Now I just have to show that $X \cong X_0$, but I am not sure whether that holds. So I am rather partial to a standard classical approach to this problem.

Any help is appreciated
Balarka
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[jvicens]

Dear Balarka,

Thank you for bringing up this interesting problem. It is indeed a fascinating question to consider whether two field extensions with isomorphic Galois groups must necessarily be equal. As you mentioned, it is well-known that there exist distinct fields over $\Bbb Q$ with isomorphic Galois groups, which suggests that the answer to this question may not be a straightforward yes or no.

In order to address this problem, it is important to understand the properties of Galois groups and how they relate to field extensions. One key concept to consider is the Galois correspondence, which states that there is a bijection between intermediate fields of a Galois extension and subgroups of the Galois group. This correspondence can provide insights into the structure of Galois groups and their relationship to field extensions.

In regards to the specific problem you posed, it is possible that two Galois extensions $K/k(t)$ and $K'/k(t)$ with isomorphic Galois groups may not be equal. This is because there could be multiple intermediate fields between $K$ and $k(t)$, and the isomorphism between their Galois groups does not necessarily imply an isomorphism between the fields themselves. However, it is also possible that the two extensions are equal, and further investigation is needed to determine if this is always the case.

Your suggestion of using non-classical Galois theory to approach this problem is also intriguing. However, as you mentioned, it may be difficult to show that $X \cong X_0$ in this context. Therefore, a combination of classical and non-classical approaches may be necessary to fully understand this problem.

In conclusion, this is a challenging and thought-provoking problem that requires further investigation. Thank you for sharing it with the scientific community and I look forward to seeing further developments on this topic.