Timeline for Examples of perfect pseudo algebraically closed fields in positive characteristic
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 14, 2015 at 12:42 | comment | added | Jason Starr | By the way, Fried-Jarden proved that every perfect pseudo algebraically closed field that contains $\overline{\mathbb{F}}_p$ (or equivalently, all roots of unity) is quasi-algebraically closed: all specializations of Fano hypersurfaces have rational points. I extended this: every specialization over such a field of a separably rationally connected variety has a rational point. Ax's conjecture says that this should be true even if the field does not contain $\overline{\mathbb{F}}_p$. | |
| Nov 13, 2015 at 13:35 | vote | accept | user45397 | ||
| Nov 13, 2015 at 3:58 | comment | added | Jason Starr | In that construction, I had the partial order wrong: $L/K < L'/K$ if there exists a $K$-embedding of $L$ in $L'$ such that $L$ is separably closed in $L'$. | |
| Nov 13, 2015 at 0:51 | comment | added | Jason Starr | You can "formally" produce such examples. Begin with any finitely generated extension $K/k$ of an algebraically closed field $k$ of characteristic $p$. Consider the filtered system of function fields $L/K$ of geometrically irreducible $K$-varieties, partially ordered by existence of embeddings $L / K < L' / K$ if there exists a $K$-embedding of $L$ in $L'$. The colimit of all of these field extensions is a field extension $E/K$ such that for every $L/E$ as above, $L$ is purely inseparable over $E$. Now replace $E$ by its perfect closure. Note: $K$ is already separably closed in $E$. | |
| Nov 12, 2015 at 23:23 | history | edited | user45397 |
edited tags
|
|
| Nov 12, 2015 at 21:50 | answer | added | user115940 | timeline score: 6 | |
| Nov 12, 2015 at 20:21 | history | asked | user45397 | CC BY-SA 3.0 |