PlumX Metrics
Embed PlumX Metrics

Explicit bounds for generators of the class group

Mathematics of Computation, ISSN: 0025-5718, Vol: 87, Issue: 313, Page: 2483-2511
2018
  • 4
    Citations
  • 0
    Usage
  • 1
    Captures
  • 0
    Mentions
  • 0
    Social Media
Metric Options:   Counts1 Year3 Year

Metrics Details

Article Description

Assuming Generalized Riemann's Hypothesis, Bach proved that the class group CℓK of a number field K may be generated using prime ideals whose norm is bounded by 12 log ΔK, and by (4 + o(1)) log ΔK asymptotically, where ΔK is the absolute value of the discriminant of K. Under the same assumption, Belabas, Diaz y Diaz and Friedman showed a way to determine a set of prime ideals that generates CℓK and which performs better than Bach's bound in computations, but which is asymptotically worse. In this paper we show that CℓK is generated by prime ideals whose norm is bounded by the minimum of 4.01 log ΔK, 4(1 + (2πe)K) log ΔK and 4( logΔK +log logΔK -(γ +log 2π)NK +1+(NK +1) log(7 logΔK)/logΔK ). Moreover, we prove explicit upper bounds for the size of the set determined by Belabas, Diaz y Diaz and Friedman's algorithms, confirming that it has size (logΔK log logΔK). In addition, we propose a different algorithm which produces a set of generators which satisfies the above mentioned bounds and in explicit computations turns out to be smaller than log Δ except for 7 out of the 31292 fields we tested.

Provide Feedback

Have ideas for a new metric? Would you like to see something else here?Let us know