Timeline for Cyclotomic extensions with split Galois group
Current License: CC BY-SA 2.5
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Dec 12, 2010 at 5:23 | comment | added | Chandan Singh Dalawat | Let $K/k$ be a finite Galois extension of number fields, and let $K'$ be the Hilbert class field of $K$. Let $e$ be the lcm of the local ramification indices for $K/k$. Wyman proves that if $e=[K\colon k]$ then the group extension $$1\rightarrow\text{Gal}(K'/K)\rightarrow\text{Gal}(K'/k)\rightarrow\text{Gal}(K/k)\rightarrow 1$$ splits. In particular, this sequence splits if $k={\bf Q}$ and $K/{\bf Q}$ is cyclic. On the other hand, he shows by a counterexample that a Galois non-cyclic extension of ${\bf Q}$ need not lead to a split extension, contrary to an assertion of C. S. Herz. (MR0337916 | |
| Nov 26, 2010 at 11:54 | history | edited | Chandan Singh Dalawat | CC BY-SA 2.5 |
TeX
|
| Nov 25, 2010 at 9:35 | answer | added | Franz Lemmermeyer | timeline score: 5 | |
| Nov 25, 2010 at 6:12 | comment | added | Jon Yard | Tom, this is a good point. So I guess this means that if E is such an extension and if s generates (Z/p), then s^{(p-1}/2} needs to map to complex conjugation in E. These sort of facts are what I was after. | |
| Nov 25, 2010 at 4:47 | comment | added | Tom Goodwillie | If n is 1 (or 2) then classifying such extensions means classifying all Galois extensions of Q. If n is 3 or 4 (or 6) then every Galois extension of Q containing the nth cyclotomic field is split by complex conjugation. | |
| Nov 25, 2010 at 3:56 | answer | added | Chandan Singh Dalawat | timeline score: 12 | |
| Nov 25, 2010 at 0:11 | history | asked | Jon Yard | CC BY-SA 2.5 |