1. EQUIDISTRIBUTION OF ALGEBRAIC NUMBERS OF NORM ONE IN QUADRATIC NUMBER FIELDS.
- Author
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PETERSEN, KATHLEEN L. and SINCLAIR, CHRISTOPHER D.
- Subjects
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ALGEBRAIC number theory , *QUADRATIC fields , *HECKE algebras , *MATHEMATICAL analysis , *MATHEMATICS , *ALGEBRA , *MATHEMATICAL mappings - Abstract
Given a fixed quadratic extension K of ℚ, we consider the distribution of elements in K of norm one (denoted $\mathscr{N}$). When K is an imaginary quadratic extension, $\mathscr{N}$ is naturally embedded in the unit circle in ℂ and we show that it is equidistributed with respect to inclusion as ordered by the absolute Weil height. By Hilbert's Theorem 90, an element in $\mathscr{N}$ can be written as $\alpha/\overline{\alpha}$ for some $\alpha \in \mathscr{O}_K$, which yields another ordering of $\mathscr{N}$ given by the minimal norm of the associated algebraic integers. When K is imaginary we also show that $\mathscr{N}$ is equidistributed in the unit circle under this norm ordering. When K is a real quadratic extension, we show that $\mathscr{N}$ is equidistributed with respect to norm, under the map β ↦ log|β|(mod log|ϵ2|) where ϵ is a fundamental unit of $\mathscr{O}_K$. [ABSTRACT FROM AUTHOR]
- Published
- 2011
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