Explicit complete residue systems in a general quadratic field.

• Suton Tadee Department of Mathematics, Faculty of Science and Technology, Thepsatri Rajabhat University, Lopburi 15000.
• Vichian Laohakosol Department of Mathematics, Faculty of Science, Kasetsart University, Bangkok 10900.
• Santad Damkaew Department of Mathematics and Computer Science, Chulalongkorn University, Bangkok 10330.

Keywords: quadratic field, complete residue system, lattice point

Abstract

Bergum explicitly determined three representations for a complete residue system in the quadratic field $\mathbb{Q(\sqrt{-3})}$ extending two earlier results in $\mathbb{Q(\sqrt{-1})}$ and $\mathbb{Q(\sqrt{-2})}$. Among these three representations, the first is simplest to derive, while the third is minimal in the sense that the sum of their absolute values is minimal. Here, we extend these results by deriving explicit representations for a complete residue system in any general quadratic field. The first representation uses lattice points in a rectangle in the first quadrant of an appropriate plane, while the second representation uses lattice points in a parallelogram, and the third representation uses lattice points in a hexagon and possesses a minimality property for imaginary quadratic fields.

References

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