Abstract
We show that q 2 divides x and p 2 divides y. Using this result, one easily proves Theorem I of chapter 1.
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Notes
- 1.
*The group H is equal to the flat cohomology group \(H^1_{\rm flat}(X,\mu_q)\). Here μ q is the group scheme of the qth roots of unity and X denotes the scheme \({\rm Spec}({\bf Z}[\zeta_p,\frac{1}{p}])\). The subgroup of \(\alpha\in H\) with the property that the extension \({\bf Q}(\zeta_p)\subset {\bf Q}(\zeta_p,{\root q\of{\alpha}})\) is unramified at the primes \({\mathfrak{q}}\) lying over q makes up the étale cohomology group \(H^1_{\rm et}(X,\mu_q)\). Since the elements α contained in the subgroup S have this property, the group S can be identified with a subgroup of \(H^1_{\rm et}(X,\mu_q)\). More precisely, it is the Selmer group
$${\rm ker}\left(H^1_{\rm et}(X,\mu_q)\longrightarrow\prod_{\mathfrak{q}|q}H^1({\rm Spec}(F_\mathfrak{q}),\mu_q)\right).$$
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© 2008 Springer-Verlag London Limited
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Schoof, R. (2008). The Double Wieferich Criterion. In: Catalan's Conjecture. Universitext. Springer, London. https://doi.org/10.1007/978-1-84800-185-5_10
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DOI: https://doi.org/10.1007/978-1-84800-185-5_10
Publisher Name: Springer, London
Print ISBN: 978-1-84800-184-8
Online ISBN: 978-1-84800-185-5
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