Places of algebraic function fields in arbitrary characteristic

Details Places of algebraic function fields in arbitrary characteristic Places of algebraic function fields in arbitrary characteristic

2010-03-29

arxiv.org/abs/1003.5686v1

Advances in Math. 188 (2004), 399-424

Mathematics

Commutative Algebra

27 pages

Scientific paper

We consider the Zariski space of all places of an algebraic function field $F|K$ 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, zero-dimensional 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 quasi-compact and that it is a spectral space.

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