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Division algebras and maximal orders for given invariants

Published online by Cambridge University Press:  26 August 2016

Gebhard Böckle  and
Damián Gvirtz
Gebhard Böckle
Affiliation:
Universität Heidelberg, Interdisziplinäres Zentrum für wissenschaftliches Rechnen (IWR), Im Neuenheimer Feld 368, 69120 Heidelberg, Germany email gebhard.boeckle@iwr.uni-heidelberg.de
Damián Gvirtz
Affiliation:
The London School of Geometry and Number Theory, Department of Mathematics, University College London, Gower Street, London WC1E 6BT, United Kingdom email damian.gvirtz.15@ucl.ac.uk

Abstract

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Brauer classes of a global field can be represented by cyclic algebras. Effective constructions of such algebras and a maximal order therein are given for $\mathbb{F}_{q}(t)$, excluding cases of wild ramification. As part of the construction, we also obtain a new description of subfields of cyclotomic function fields.

Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 19 , Special Issue A: Algorithmic Number Theory Symposium XII , 2016 , pp. 178 - 195
Copyright
© The Author(s) 2016 

References

Anglès, B., ‘Bases normales relatives en caractéristique positive’, J. Théor. Nombres Bordeaux 14 (2002) no. 1, 1–17.Google Scholar
Böckle, G. and Butenuth, R., ‘On computing quaternion quotient graphs for function fields’, J. Théor. Nombres Bordeaux 24 (2012) no. 1, 73–99.CrossRefGoogle Scholar
Chapman, R. J., ‘Carlitz modules and normal integral bases’, J. Lond. Math. Soc. (2) 44 (1991) no. 2, 250–260.Google Scholar
Deuring, M., Algebren , 2nd edn, Ergebnisse der Mathematik und ihrer Grenzgebiete 41 (Springer, Berlin, 1968).CrossRefGoogle Scholar
Galovich, S. and Rosen, M., ‘Units and class groups in cyclotomic function fields’, J. Number Theory 14 (1982) no. 2, 156–184.Google Scholar
Goss, D., Basic structures of function field arithmetic , Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 35 (Springer, Berlin, 1996).Google Scholar
Guruswami, V., ‘Cyclotomic function fields, Artin–Frobenius automorphisms, and list error correction with optimal rate’, Algebra Number Theory 4 (2010) no. 4, 433–463.Google Scholar
Jantzen, J. C. and Schwermer, J., Algebra (Springer, Berlin, 2006).Google Scholar
Knuth, D. E., The art of computer programming, Vol. 1: Fundamental algorithms , 3rd edn (Addison-Wesley, Reading, MA, 1997).Google Scholar
Lettl, G., ‘The ring of integers of an abelian number field’, J. reine angew. Math. 404 (1990) 162–170.Google Scholar
Lorenz, F., ‘Fields with structure, algebras and advanced topics’, Algebra, Vol. II , Universitext (Springer, New York, 2008). Translated from the German by Silvio Levy, with the collaboration of Levy.Google Scholar
Neukirch, J., Algebraische Zahlentheorie , 1st edn (Springer, Berlin, 1992).Google Scholar
Reiner, I., Maximal orders (Academic Press, London, 1975).Google Scholar
Rosen, M., Number theory in function fields , Graduate Texts in Mathematics 210 (Springer, New York, 2002).Google Scholar
Rzedowski-Calderón, M. and Villa-Salvador, G., ‘Conductor-discriminant formula for global function fields’, Int. J. Algebra 5 (2011) no. 29–32, 1557–1565.Google Scholar
Schwinning, N., ‘Ein Algorithmus zur Berechnung von Divisionsalgebren über $\mathbb{Q}$ zu vorgegebenen Invarianten’, Diplomarbeit, Universität Duisburg-Essen, Germany, 2011.Google Scholar
Serre, J.-P., Local fields, Graduate Texts in Mathematics 67 (Springer, New York, 1979). Translated from the French by Marvin Jay Greenberg.Google Scholar