Timeline for number fields with no unramified extensions?
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Aug 8, 2013 at 2:14 | comment | added | Filippo Alberto Edoardo | I am sorry for a vary late comment on Mahesh' one, but recently Morisawa (student of Komatsu, I think) has shown that given a finite set of primes $S$, for each number field $F$ inside $\mathbb{Q}^{cycl,S}=\prod_{p\in S}\mathbb{Q}^{cycl,p}$, there is a constant $c=c(S,F)$ such that each prime $\ell>c$ whose decomposition field is $F$ does not divide the class number of any other number field contained in $\mathbb{Q}^{cycl,S}$. It appeared in J. Nmber Theory 133 (2013). | |
| Oct 6, 2010 at 14:20 | comment | added | Mahesh Kakde | No. At this moment it is fair to say that Coates raised them as questions based on some numerical evidence though he called them conjectures. There is a very weak numerical evidence for the second part. Some japanese mathematicians (Okazaki, Fukuda, Komatsu were the names mentioned in the talk of Coates) have shown that for p=2,3 the ideal class groups of fields in the $\mathbb{Z}_p$ extension of $\mathbb{Q}$ has no prime divisor less than a million (I guess). | |
| Oct 6, 2010 at 12:28 | comment | added | Cam McLeman | Interesting, thanks. Do you know of a written reference somewhere? | |
| Oct 6, 2010 at 11:36 | comment | added | Mahesh Kakde | Since you mentioned about the problem of infinitely many fields of class number one, I would like to mention the conjectures that Coates recently made in a talk in Kyoto. Take the extension of $\mathbb{Q}$ with Galois group $\widehat{\mathbb{Z}}$. Then he conjectures that the set class numbers of all fields in this $\widehat{\mathbb{Z}}$ extension is a bounded set. He also conjectures that for any prime $p$ the class number of all fields in the $\mathbb{Z}_p$ extension of $\mathbb{Q}$ is 1. This was conjectured by Weber for $p=2$. | |
| Oct 6, 2010 at 11:22 | history | edited | Cam McLeman | CC BY-SA 2.5 |
added 84 characters in body
|
| Oct 6, 2010 at 3:53 | comment | added | Mahesh Kakde | There is more discussion about this here mathoverflow.net/questions/31538/… | |
| Oct 6, 2010 at 2:40 | history | answered | Cam McLeman | CC BY-SA 2.5 |