Journal of the Ramanujan Mathematical Society

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The Fundamental Unit of Some Quadratic, Cubic or Quartic Orders


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1 Institut de Mathematiques de Luminy, UMR 6206, 163, Avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France
     

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We explain why it is reasonable to conjecture that if ∈ is a totally imaginary quartic unit, then ∈ is in general a fundamental unit of the quartic order Z[∈], order whose group of units is of rank equal to one. We partially prove this conjecture. This generalizes a result of T. Nagell, who proved in 1930 a similar result for real cubic units with two non real conjugates.
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  • The Fundamental Unit of Some Quadratic, Cubic or Quartic Orders

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Authors

Stephane R. Louboutin
Institut de Mathematiques de Luminy, UMR 6206, 163, Avenue de Luminy, Case 907, 13288 Marseille Cedex 9, France

Abstract


We explain why it is reasonable to conjecture that if ∈ is a totally imaginary quartic unit, then ∈ is in general a fundamental unit of the quartic order Z[∈], order whose group of units is of rank equal to one. We partially prove this conjecture. This generalizes a result of T. Nagell, who proved in 1930 a similar result for real cubic units with two non real conjugates.