 Cyclotomic Function Fields with Many Rational PlacesAlice Keller ()
A chapter in Finite Fields and Applications, 2001, pp 293-302 from Springer Abstract: Abstract Let $$ A = \mathbb{F}{_{q}}[T] $$ be the polynomial ring in the variable T and $$ K = \mathbb{F}{_{q}}[T] $$ the rational function field over $$ \mathbb{F}{_{q}} $$ (the finite field with q elements), and let K ∞ be the completion of K at the place $$ \infty : = \frac{1}{T} $$ . Furthermore let C be the completion of a fixed algebraic closure of K ∞. We aim to construct extensions K ⊂ K′ ⊂ C with many rational places relative to the genus g(K′) of K′. As a first step we consider the cyclotomic fields K(n)/K with n ∈ A, which are generated analogously to the classical cyclotomic fields over ℚ. Then we consider certain decomposition fields and their intersections. Here we know a lower bound for the number of rational places. We get explicit formulas to calculate the genus, but they depend on the relative position of some subgroups of the multiplicative group (A/(n))* of the ring A/(n). So the concrete calculation of examples must be done by computer. With a special program we made a systematic search for q = 2 and found three improved lower bounds. Keywords: Galois Group; Algebraic Closure; Integral Closure; Rational Place; Decomposition Group (search for similar items in EconPapers) There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-642-56755-1_22 Ordering information: This item can be ordered from DOI: 10.1007/978-3-642-56755-1_22 Access Statistics for this chapter More chapters in Springer Books from Springer |
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