Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question
Have you tried using the hint? What is it suggesting? – Qiaochu Yuan May 21 '11 at 9:23

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?

share|cite|improve this answer
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

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.