91Ó°ÊÓ

Show that if \(E\) is a separable extension of \(F\) of degree \([E: F]=2\), then \(\operatorname{Gal}(E / F) \approx \mathbb{Z}_{2}\)

Show that if \(\alpha\) is a real number algebraic over \(\mathbb{Q}\) of degree \(4,\) then \(\alpha\) is not constructible by straightedge and compass if and only if the Galois group of the splitting field of its minimal polynomial over \(\mathbb{Q}\) is isomorphic to \(A_{4}\) or \(S_{4}\).

Let \(F\) be a field of characteristic \(p\). Show that \(F\) is a perfect field if and only if for every element \(a \in F\) there is an element \(b \in F\) such that \(a=b^{p}\).

Let \(\alpha\) be a positive real number that is constructible by straightedge and compass, and \(f(x)\) its minimal polynomial over \(Q\). Show that \(f(x)\) is solvable by radicals over \(Q\)