91Ó°ÊÓ

Let \(E=Q(\sqrt{2}, \sqrt{3}, i),\) and consider the intermediate fields \(\mathbb{Q}\) \subseteq \(K \subseteq E\). $$ \text { For each } K, \text { calculate }|\operatorname{Gal}(E / K)| \text { . } $$

In Exercises 1 through 7 determine for the indicated \(n\) whether or not the regular \(n\) -gon is constructible by straightedge and compass. $$ n=52 $$

In Exercises 1 through 8 express the splitting field of the indicated polynomial \(f(x) \in \mathbb{Q}[x]\) as a radical extension of \(\mathrm{Q}\). $$ x^{4}+2 x^{2}+1 $$

In Exercises 1 through 7 determine for the indicated \(n\) whether or not the regular \(n\) -gon is constructible by straightedge and compass. $$ n=56 $$

Let \(E=Q(\sqrt{2}, \sqrt{3}, i),\) and consider the intermediate fields \(\mathbb{Q}\) \subseteq \(K \subseteq E\). $$ \text { For each } K, \text { calculate }|\chi(K)| \text { . } $$

In Exercises 1 through 8 express the splitting field of the indicated polynomial \(f(x) \in \mathbb{Q}[x]\) as a radical extension of \(\mathrm{Q}\). $$ x^{3}+x^{2}+x+1 $$

Calculate the \(\mathrm{Gal}(E / \mathrm{Q})\). where \(E\) is the splitting field in \(\mathrm{C}\) of the indicated polynomial \(f(x) \in \mathbb{Q}[x]\). $$ f(x)=x^{4}-2 x $$

In Exercises 8 through 13 determine for the indicated \(n\) whether or not the regular \(n\) -gon is constructible by marked ruler and compass. $$ n=12 $$

In Exercises 1 through 8 express the splitting field of the indicated polynomial \(f(x) \in \mathbb{Q}[x]\) as a radical extension of \(\mathrm{Q}\). $$ x^{4}+x^{3}+2 x^{2}+x+1 $$

Let \(E=Q(\sqrt{2}, \sqrt{3}, i),\) and consider the intermediate fields \(\mathbb{Q}\) \subseteq \(K \subseteq E\). $$ \text { Describe } \operatorname{Gal}(E / Q) $$