scholarly journals HILBERT 3-CLASS FIELD TOWERS OF IMAGINARY CUBIC FUNCTION FIELDS

2013 ◽  
Vol 26 (1) ◽  
pp. 161-166 ◽  
Author(s):  
Hwanyup Jung
Keyword(s):  
Function Fields ◽  
Class Field ◽  
Class Field Towers ◽  
Cubic Function Fields ◽  
Cubic Function

scholarly journals Tabulation of cubic function fields via polynomial binary cubic forms

2012 ◽  
Vol 81 (280) ◽  
pp. 2335-2359 ◽  
Author(s):  
Pieter Rozenhart ◽  
Michael Jacobson ◽  
Renate Scheidler
Keyword(s):  
Function Fields ◽  
Cubic Forms ◽  
Cubic Function Fields ◽  
Cubic Function

An Explicit Treatment of Cubic Function Fields with Applications

2010 ◽  
Vol 62 (4) ◽  
pp. 787-807 ◽  
Author(s):  
E. Landquist ◽  
P. Rozenhart ◽  
R. Scheidler ◽  
J. Webster ◽  
Q. Wu
Keyword(s):  
Function Fields ◽  
Standard Form ◽  
Irreducible Polynomial ◽  
Zeta Functions ◽  
Cube Root ◽  
Integral Bases ◽  
Root Extraction ◽  
Necessary And Sufficient Criteria ◽  
Cubic Function Fields ◽  
Cubic Function

AbstractWe give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few square-free polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields.

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