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  • 1 Institute for Analysis and Computational Number Theory, Graz University of Technology, Steyrergasse 30/IV, A-8010 Graz, Austria
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Abstract  

Let m ≠ 0 be an integer which is not a perfect square and consider number fields of the form

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{Q}\left[ {\sqrt[4]{m}} \right]$$ \end{document}
. We characterize all orders of the form
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{Z}\left[ {\sqrt[4]{m}} \right]$$ \end{document}
which admit a unit power integral basis, i.e., there exists a unit ε such that 1, ε, ε2 and ε3 is an integral basis of
\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\mathbb{Z}\left[ {\sqrt[4]{m}} \right]$$ \end{document}
.