On principal ideal testing in totally complex quartic fields and the determination of certain cyclotomic constants

Authors:
Johannes Buchmann and H. C. Williams

Journal:
Math. Comp. **48** (1987), 55-66

MSC:
Primary 11Y40; Secondary 11R16

DOI:
https://doi.org/10.1090/S0025-5718-1987-0866098-3

MathSciNet review:
866098

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Abstract | References | Similar Articles | Additional Information

Abstract: Let be any totally complex quartic field. Two algorithms are described for determining whether or not any given ideal in is principal. One of these algorithms is very efficient in practice, but its complexity is difficult to analyze; the other algorithm is computationally more elaborate but, in this case, a complexity analysis can be provided.

These ideas are applied to the problem of determining the cyclotomic numbers of order 5 for a prime . Given any quadratic (or quintic) nonresidue of *p*, it is shown that these cyclotomic numbers can be efficiently computed in binary operations.

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DOI:
https://doi.org/10.1090/S0025-5718-1987-0866098-3

Article copyright:
© Copyright 1987
American Mathematical Society