# zbMATH — the first resource for mathematics

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
##### Keywords:
Fermat’s Last Theorem; exponent five
Full Text: