## Galois group of the Q(gamma_1,…) where Zeta(1/2+i*gamma_k)=0

Hello and happy new year.

it seems quite strange to me, but I couldn't find any question on MO about the Galois group mentioned in the title. So, letting $s=1/2+i*\gamma_k$ be the $k$-th non-trivial zero of Zeta of positive imaginary part, have some results towards the determination of the structure of $Gal(Q(\gamma_1,\gamma_2,...)/Q)$ been obtained so far? Thank you in advance.

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It doesn't seem strange to me that nobody has asked such a question, since almost certainly all those $\gamma_i$ are transcendental and probably algebraically independent over the rationals. So the field you write down is likely to be isom. to the rationals adjoined countably many indeterminates. There is no proof that any of those $\gamma_i$ are transcendental so nothing is known. In the same spirit, it's not strange that nobody has asked about the rationals adjoined Euler's constant. – KConrad Jan 1 2012 at 17:30