Abstract
It turns out that the group of fractional ideals \(\mathbb{I}_{F}\) is not an interesting invariant of the quadratic field F: for different fields F, F ′, Exer. 5.1.7 shows that \(\mathbb{I}_{F}\mathop{\cong}\mathbb{I}_{F^{\prime}}\). To get an object which does reflect the arithmetic of F, we consider a quotient of \(\mathbb{I}_{F}\).
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Further Reading
Harper, M.: \(\mathbb{Z}[\sqrt{14}]\) is Euclidean. Canad. J. Math. 56(1), 55–70 (2004)
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Trifković, M. (2013). The Ideal Class Group and the Geometry of Numbers. In: Algebraic Theory of Quadratic Numbers. Universitext. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7717-4_5
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DOI: https://doi.org/10.1007/978-1-4614-7717-4_5
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