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On Towers and Composita of Towers of Function Fields over Finite Fields

✍ Scribed by Arnaldo Garcia; Henning Stichtenoth; Michael Thomas

Publisher
Elsevier Science
Year
1997
Tongue
English
Weight
296 KB
Volume
3
Category
Article
ISSN
1071-5797

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✦ Synopsis

For a tower F 1 Κ• F 2 Κ• ΠΈ ΠΈ ΠΈ of algebraic function fields F i ‫ކ/‬ q , define Ο­ lim iǞȍ N(F i )/g(F i ), where N(F i ) is the number of rational places and g(F i ) is the genus of F i ‫ކ/‬ q . The tower is said to be asymptotically good if ΟΎ 0. We give a very simple explicit example of an asymptotically good tower for all non-prime fields ‫ކ‬ q . In this example, all steps F iΟ©1 /F i are tamely ramified Kummer extensions. We then show that any function field F/‫ކ‬ q having at least one rational place can be embedded into an asymptotically good tower, and we study the behaviour of in the compositum of a tower F 1 Κ• F 2 Κ• ΠΈ ΠΈ ΠΈ with an extension E/F 1 .

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