Places of algebraic function fields in arbitrary characteristic
Abstract
We consider the Zariski space of all places of an algebraic function field $FK$ of arbitrary characteristic and investigate its structure by means of its patch topology. We show that certain sets of places with nice properties (e.g., prime divisors, places of maximal rank, zerodimensional discrete places) lie dense in this topology. Further, we give several equivalent characterizations of fields that are large, in the sense of F. Pop's Annals paper {\it Embedding problems over large fields}. We also study the question whether a field $K$ is existentially closed in an extension field $L$ if $L$ admits a $K$rational place. In the appendix, we prove the fact that the Zariski space with the Zariski topology is quasicompact and that it is a spectral space.
 Publication:

arXiv eprints
 Pub Date:
 March 2010
 arXiv:
 arXiv:1003.5686
 Bibcode:
 2010arXiv1003.5686K
 Keywords:

 Mathematics  Commutative Algebra;
 12J10
 EPrint:
 27 pages