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Dear all, I would be grateful if someone could provide a solution to the following problem (using decomposition and inertia groups):

Find a finite extension of $\mathbb{Q}$ in which all primes split.

[Hint: Use the fact that a prime splits if and only if its decomposition group is not the full Galois group (and that the decomposition group is cyclic for all unramified primes)]

Many thanks, Mohammad.

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Have you tried using the hint? What is it suggesting? –  Qiaochu Yuan May 21 '11 at 9:23
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It may be possible to do this without using the hint. The polynomial $(x^2-13)(x^2-17)(x^2-221)$ has a root mod $p$ for all $p$; does that imply that every $p$ splits in ${\bf Q}(\sqrt{13},\sqrt{17})$? Or do I have my wires crossed?

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The more I think about my answer, the less I like it. The polynomial has a root mod $5$, but $5$ is not a quadratic residue to any of the moduli $13$, $17$, or $221$. So, does $5$ split in the field? –  Gerry Myerson May 22 '11 at 0:28
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