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In this paper, we show the existence of a non-solvable Galois extension of$\Q$ which is unramified outside 2. The extension $K$ we construct has degree$2251731094732800=2^{19}(3\cdot 5\cdot 17\cdot 257)^2$ and has rootdiscriminant $\delta_K <2^{{47/8}}=58.68...$, and is totally complex.

A non-solvable Galois extension of $\Q$ ramified at 2 only