91Ó°ÊÓ

In Exercises 1 through 3 express the indicated symmetric functions of \(n\) indeterminates in terms of the elementary symmetric functions in \(n\) indeterminates. $$ n=2 \quad\left(x_{1}-x_{2}\right)^{2} $$

Calculate the Galois group \(\mathrm{Gal}(E / Q)\) for the indicated fields \(E\) $$ E=Q(\sqrt{3}, \sqrt{5}) $$

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

In Exercises 1 through 4 determine whether the indicated field extension is a Galois extension. $$ Q(\sqrt{3} i) \text { over } Q $$

In Exercises 1 through 7 determine for the indicated \(n\) whether or not the regular \(n\) -gon is constructible by straightedge and compass. $$ n=22 $$

Calculate the Galois group \(\mathrm{Gal}(E / Q)\) for the indicated fields \(E\) \(E=\mathbb{Q}(\omega),\) where \(\omega\) is the primitive cube root of unity

In Exercises 1 through 8 express the splitting field of the indicated polynomial \(f(x) \in \mathbb{Q}[x]\) as a radical extension of \(\mathrm{Q}\). $$ x^{3}-5 $$

Determine whether the indicated field extension is a Galois extension. $$ Q(\sqrt[3]{3}) \text { over } Q $$

Express the indicated symmetric functions of \(n\) indeterminates in terms of the elementary symmetric functions in \(n\) indeterminates. $$ n=3 \quad\left(x_{1}-x_{2}\right)^{2}\left(x_{1}-x_{3}\right)^{2}\left(x_{2}-x_{3}\right)^{2} $$

Express the indicated symmetric functions of \(n\) indeterminates in terms of the elementary symmetric functions in \(n\) indeterminates. $$ n=3 \quad\left(x_{1}+x_{2}\right)\left(x_{1}+x_{3}\right)\left(x_{2}+x_{3}\right) $$