Ample field

2020 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