91Ó°ÊÓ

L.t \(F\) be a finite extension of the rationals. Show that there is only a finite number of roots of unity in \(F\).

Describe the splitting field of \(t^{5}-7\) over the rationals. What is its degree? Show that the Galois group is generated by two elements \(\sigma\), t satisiying the relation $$ \sigma^{5}=1, \quad \tau^{4}=1, \quad \operatorname{tat}^{-1}=a^{2} $$