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Finding eisenstein elements in cyclic number fields of odd prime degree

Published online by Cambridge University Press:  17 April 2009

Vincenzo Acciaro
Vincenzo Acciaro
Affiliation:
School of Computer Science, Carleton University, Ottawa, Ont, Kl.S 5B6, Canada
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Abstract

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Let L = Q[α] be a cyclic number field of odd prime degree q over the field Q of rationals. In this paper we give an algorithm to compute the discriminant of L/Q, which relies upon a fast method to find Eisenstein elements in L. The algorithm accepts as input the minimal polynomial of α over Q and a complete factorisation of the discriminant of α, and computes, in time polynomial in the size of the input, a list consisting of all the ramified primes with corresponding Eisenstein elements.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 52 , Issue 2 , October 1995 , pp. 331 - 340
Copyright
Copyright © Australian Mathematical Society 1995

References

[1]Acciaro, V., ‘Solvability of norm equations over Abelian number fields of prime degree’, (manuscript, 1994).Google Scholar
[2]Cohen, H., A course in computational algebraic number theory (Springer-Verlag, Berlin, Heidelberg, New York, 1993).CrossRefGoogle Scholar
[3]Cox, D.A., Primes of the form x 2 + ny 2 (John Wiley and Sons, New York, 1989).Google Scholar
[4]Goldstein, L.J., Analytic number theory (Prentice-Hall, Englewood Cliffs, New Jersey, 1971).Google Scholar
[5]Lang, S., Algebraic number theory (Addison-Wesley, Reading, Massachusetts, 1970).Google Scholar
[6]Pohst, M.E., ‘Three principal tasks of computational algebraic number theory’, in Number theory and applications, Proc. NATO Advanced Study Inst. (Kluwer Academic Publisher, 1989), pp. 123–133.Google Scholar
[7]Urazbaev, B.M., ‘On the discriminant of a cyclic field of prime degree’, Izv. Akad. Nauk Kazah. SSR Math. Meh. 4 (1950), 19–32.Google Scholar
[8]Weiss, E., Algebraic number theory (McGraw-Hill, New York, 1963).Google Scholar
[9]Zassenhaus, H., ‘Ein Algorithmus zur Berechnung einer Minimalbasis über gegebener Ordnung’, Funktsional Anal. (1967), 90–103.Google Scholar