# Is ramification of number fields first order?

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 $K/\mathbb{Q}$?

-
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 math.uconn.edu/~kconrad/math5230f08/schoof.pdf 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
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 1, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 0, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0, "katex": 0, "math-container": 0, "wp-katex-eq": 0, "align": 0, "equation": 0, "x-ck12": 0, "texerror": 0, "math_score": 0.9089366793632507, "perplexity": 177.85705459450486}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "plain"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "warc_path": "s3://commoncrawl/crawl-data/CC-MAIN-2016-26/segments/1466783395160.19/warc/CC-MAIN-20160624154955-00131-ip-10-164-35-72.ec2.internal.warc.gz"}