Timeline for Orders in number fields
Current License: CC BY-SA 3.0
17 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| May 16, 2014 at 2:58 | history | edited | Ramin | CC BY-SA 3.0 |
Updated the status of the project that inspired the question.
|
| May 29, 2013 at 15:06 | comment | added | Ramin | Thanks everyone. You can actually count the number of subrings of index $p$. Namely, if $r(x) \mod p$ has $u$ factors of degree $1$ and $w$ factors of degree $2$, and the rest of higher degrees, then the number of subrings of ${\mathbb Z}/p{\mathbb Z}[x]/(r(x))$ of index $p$ is ${u \choose 2} + w$. | |
| May 28, 2013 at 1:22 | history | bounty ended | Ramin | ||
| May 25, 2013 at 16:41 | comment | added | anon | The $r=1$ case can be handled as follows. Each order in $R$ of index $p$ is uniquely determined by its image in $R/pR$. Now $R/pR$ is an \'etale $k := \mathbf{F_p}$-algebra of rank $n$ as $p$ is unramified. Now the set of such rings injects into the set of index $p$ $k$-subalgebras of $R/pR \otimes_k \overline{k} \simeq \overline{k}^n$. The latter set is a combinatorial object (it's the set of surjective maps from an $n$-element set to an $(n-1)$-element set), and can be bounded above by the set of index $p$ subrings of $\mathbf{Z}^n$. | |
| May 21, 2013 at 23:41 | comment | added | Frank Thorne | I found a paper by Melanie Wood which might be relevant, although I did not check it carefully: arxiv.org/pdf/1007.5508v1.pdf | |
| May 21, 2013 at 23:38 | comment | added | Frank Thorne | Have you tried (probably you have) writing down a binary $n$-ic form and playing the same game as in Theorem 9 to Lemma 13 of Bhargava, Shankar, and Tsimerman's paper arxiv.org/pdf/1005.0672.pdf ? This doesn't "just work", the details are much messier, but since you need something weaker I wonder if it is possible to salvage the result you are looking for? | |
| May 21, 2013 at 18:58 | comment | added | Ramin | In our work we only need this for unramified primes. | |
| May 21, 2013 at 2:31 | comment | added | Will Sawin | It is clear that this question can be studied locally at $p$. In other words, this depends only on $R \otimes \mathbb Z_p$, so it depends only on $K \otimes \mathbb Q_p$, which is just a product of $p$-adic fields. | |
| May 21, 2013 at 2:05 | comment | added | Will Sawin | Is it obvious that the elementary reformulation is the same, given that one is restricted to $p$ unramified and the other is not? | |
| May 21, 2013 at 1:09 | history | bounty started | Ramin | ||
| May 15, 2013 at 17:08 | history | edited | Ramin | CC BY-SA 3.0 |
added 17 characters in body
|
| May 15, 2013 at 2:02 | history | edited | Ramin | CC BY-SA 3.0 |
added 330 characters in body
|
| May 8, 2013 at 19:39 | history | edited | Ramin | CC BY-SA 3.0 |
added 288 characters in body
|
| May 8, 2013 at 17:05 | comment | added | Marc Palm | Who voted to close? And why? | |
| May 8, 2013 at 16:19 | history | edited | Ramin |
edited tags
|
|
| May 8, 2013 at 15:59 | history | edited | Dima Pasechnik | CC BY-SA 3.0 |
fixed the link
|
| May 8, 2013 at 15:50 | history | asked | Ramin | CC BY-SA 3.0 |