Difference between revisions of "Ample field"
From Encyclopedia of Mathematics
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| − | * Moshe Jarden, "Algebraic patching", Springer (2011) ISBN 978-3-642-15127-9 {{ZBL|1235.12002}} | + | * 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}} | * 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}} | ||
Latest revision as of 17:39, 12 November 2023
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
How to Cite This Entry:
Ample field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ample_field&oldid=51412
Ample field. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Ample_field&oldid=51412