View source for Ample field

{{TEX|done}}{{MSC|12E30|11R58,12F12}}

A field which is [[existentially closed]] in its field of [[formal power series]].  Examples include [[pseudo algebraically closed field]]s, [[real closed field]]s and [[Henselian]] fields.

A field $K$ is ample if and only if every absolutely irreducible curve over $K$ with a simple $K$-point has infinitely many $K$-points.

If $K$ is ample, then the [[Galois theory, inverse problem of|inverse Galois problem]] for $K(T)$ is solved: every finite group occurs as a Galois group over $K(T)$.

A field is '''virtually ample''' if some finite extension is ample.

====References====
* Moshe Jarden, "Algebraic patching", Springer (2011) {{ISBN|978-3-642-15127-9}} {{ZBL|1235.12002}}
* Pierre Dèbes, Bruno Deschamps, "The regular inverse Galois problem over large fields" ''in ''Schneps, Leila (ed.) et al., Geometric Galois actions '''2'''", LMS Lecture Notes '''243''' Cambridge (1997) pp119-138 {{ZBL|0905.12004}}