# Ample field

From Encyclopedia of Mathematics

2010 Mathematics Subject Classification: *Primary:* 12E30 *Secondary:* 11R5812F12 [MSN][ZBL]

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

**How to Cite This Entry:**

Ample field.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Ample_field&oldid=51412