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The Modularity Theorem

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Quadratic Number Fields

Part of the book series: Springer Undergraduate Mathematics Series ((SUMS))

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Abstract

This chapter presents the quadratic reciprocity law and the modularity of Kronecker characters and applies the results to Terjanian’s theorem on the Fermat equation of exponent 2p.

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Notes

  1. 1.

    Beginners in mathematics may find it hard to believe that mathematicians think of finite fields (and even p-adic numbers) as being simpler objects than integers. One possible way of measuring the simplicity of structures A and B is counting homomorphisms from A and B into structures C. For example, there are many homomorphisms from \({\mathbb Z}\) to finite fields \({\mathbb F}_p\), whereas the only homomorphisms from \({\mathbb F}_p\) to \({\mathbb Z}\) or to finite fields are either the trivial homomorphism mapping everything to 0 or (in the case of \({\mathbb F}_p \longrightarrow {\mathbb F}_p\)) an isomorphism.

  2. 2.

    See [136] and [41]; our presentation is a simplification of the one given in [62].

  3. 3.

    This proof is essentially due to Frobenius [41].

  4. 4.

    See, e.g., [36].

  5. 5.

    This proof is lifted from Davenports beautiful book [27]; its basic idea goes back to the proof given by Arnold Scholz in [112].

  6. 6.

    See the beautiful article [24] by Cosgrave and Dilcher for an introduction to such congruences.

  7. 7.

    There is a misprint in Terjanian [121, Equation (2)].

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  1. Jagstzell, Germany

    Franz Lemmermeyer

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  1. Franz Lemmermeyer
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Lemmermeyer, F. (2021). The Modularity Theorem. In: Quadratic Number Fields. Springer Undergraduate Mathematics Series. Springer, Cham. https://doi.org/10.1007/978-3-030-78652-6_3

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