As a consequence of a theorem of Rost-Springer, we establish that the cyclicity problem for central simple algebra of degree~5 on fields containg a fifth root of unity is equivalent to the study of anisotropic elements of order 5 in the split group of type~$E_8$.

Keywords:

algèbres simples centrales, cohomologie galoisienne algèbres simples centrales, cohomologie galoisienne

MSC Classifications:

16S35, 12G05, 20G15 show english descriptions Twisted and skew group rings, crossed products
Galois cohomology [See also 14F22, 16Hxx, 16K50]
Linear algebraic groups over arbitrary fields 16S35 - Twisted and skew group rings, crossed products
12G05 - Galois cohomology [See also 14F22, 16Hxx, 16K50]
20G15 - Linear algebraic groups over arbitrary fields

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