MathOverflow is a question and answer site for professional mathematicians. 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

Fix a prime number $p$. Is there a first order sentence $\phi_p$ in the language of fields such that $\phi_p$ holds in a number field $K$ if and only if the prime $p$ is unramified in the field extension $K/\mathbb{Q}$?

share|cite|improve this question
Could you provide a bit more explanation for those of us who are not number-theory experts? – Joel David Hamkins Mar 20 '12 at 14:43
Hi Joel. There are many intuitions about what ramification means, but here is the easiest definition. Let $O_K$ be the integral closure of $\mathbb{Z}$ in $K$. Then, since $O_K$ is a Dedekind domain, you can write $pO_K$ as a product $\mathfrak{p}_1^{e_1}...\mathfrak{p_n}^{e_n}$ where the $\mathfrak{p_i}$'s are different prime ideals of $O_K$. The prime $p$ is said to be unramified if $e_1=...=e_n=1$ (i.e. if no prime ideal appears more than once in the decomposition of $pO_K$). The most intuitive kind of ramification is by taking roots of $p$ (e.g. $\mathbb{Q}(\sqrt{2})$ ramifies at $2$), but – Makhalan Duff Mar 20 '12 at 14:49
ramification can appear in more subtle ways, like this: the prime $2$ ramifies in $\mathbb{Q}(i)$. – Makhalan Duff Mar 20 '12 at 14:49
(in scheme theory, this number-theoretic notion of ramification becomes the same as the geometric notion of ramification of maps between varieties.) – Makhalan Duff Mar 20 '12 at 14:53
I am far from an expert in logic, but my gut feeling is that the algorithm on p.28 of the pdf here could yield a first order statement because $p$ ramifies in $K$ if and only if $p$ divides the discriminant of $\mathcal O_K$ as a $\mathbb Z$-algebra. – stankewicz Mar 20 '12 at 17:08

I THINK, the answer is YES by a paper of Robert Rumely:

Undecidability and definability of the theory of global fields, 1980,

where he showed that "a great variety of number-theoretic objects, from rings of integers and valuations, to zeta-functions and adele rings" are definable. Many of them are actually uniformally definable.

For the ramification issue, see the bottom of Page 210.

share|cite|improve this answer

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.