Abstract
This chapter presents the arithmetic of ideals and the class group of quadratic number fields; as an application, we solve equations of Bachet–Mordell type.
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Notes
- 1.
This lemma is related to Dedekind’s “Prague Theorem”; see [80]. At this point we are using the fact that the ring \(\mathcal O_k\) is integrally closed, i.e., is equal to the maximal order.
- 2.
In a similar way it can be shown that the Fermat equation x p + y p = z p for prime exponents p has only trivial solutions with xyz = 0 if p does not divide the class number of the field \({\mathbb Q}(\zeta _p)\) of p-th roots of unity—this is essentially Kummer’s approach to Fermat’s Last Theorem.
- 3.
This is the 3-Sylow subgroup of the ideal class group, which consists of all ideal classes whose order is a power of 3.
References
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Lemmermeyer, F. (2021). Ideals in Quadratic Number Fields. In: Quadratic Number Fields. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-030-78652-6_6
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