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Über den letzten Fermatschen Satz für \(n = 5\). (Czech) JFM 41.0249.18
Čas. Mat. Fys. 39, 185-195 (1910); 39, 305-317 (Bohemian) (1910).
In the first part of this article the author proves that the ring \(\mathbb Z[\zeta]\) of fifth roots of unity is Euclidean with respect to the norm. In the second part he shows that its unit group is generated by \(\zeta\) and \(\frac{1+\sqrt{5}}2\). As an application, he shows that equations of the form \(\alpha^5 + \beta^5 = \eta \gamma^5\) for units \(\eta \in \mathbb Z[\zeta]\) do not have nontrivial solutions in \(\mathbb Z[\zeta]\). The method is the well known infinite descent already employed by Kummer in the general case.
MSC:
11D41 Higher degree equations; Fermat’s equation
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