91Ó°ÊÓ

Show that \(x^{4}+1\) is irreducible in \(Q[x]\), as we asserted in Example \(54.7 .\)

Describe the group of the polynomial \(\left(x^{5}-2\right) \in(Q(\zeta))[x]\) over \(Q(\zeta)\), where \(\zeta\) is a primitive 5 th root of unity.

In the easiest way possible, describe the group of the polynomial $$ \left(x^{8}-1\right) \in \mathbb{Q}[x] $$ over Q.

Find the splitting field \(K\) in \(C\) of the polynomial \(\left(x^{4}-4 x^{2}-1\right) \in Q[x]\). Compute the group of the polynomial over \(\mathrm{Q}\) and exhibit the correspondence between the subgroups of \(G(K / Q)\) and the intermediate fields, In other words, do the complete job.

Express each of the following symmetric functions in \(y_{1} \cdot y_{2}, y_{3}\) over \(\mathbb{Q}\) as a rational function of the elementary symmetric functions \(s_{1}, s_{2}, s_{3}\). a. \(y_{1}{ }^{2}+y_{2}^{2}+y_{3}{ }^{2}\) b. \(\frac{y_{1}}{y_{2}}+\frac{y_{2}}{y_{1}}+\frac{y_{1}}{y_{3}}+\frac{y_{3}}{y_{1}}+\frac{y_{2}}{y_{3}}+\frac{y_{3}}{y_{2}}\)

Let \(\alpha_{1}, \alpha_{2}, \alpha_{3}\) be the zeors in \(C\) of the polynomial $$ \left(x^{3}-4 x^{2}+6 x-2\right) \in Q[x] $$ Find the polynomial having as zeros precisely the following: a. \(\alpha_{1}+\alpha_{2}+\alpha_{3}\) b. \(\alpha_{1}^{2}, \alpha_{2}^{2}, \alpha_{3}^{2}\)

Let \(f(x) \in F[x]\) be a monic polynomial of degree \(n\) having all its irreducible factors separable over \(F\). Let \(K \leq \bar{F}\) be the splitting field of \(f(x)\) over \(F\), and suppose that \(f(x)\) factors in \(K[x]\) into $$ \prod_{i=1}^{n}\left(x-\alpha_{i}\right) $$ Let $$ \Delta(f)=\prod_{i

An element of \(C\) is an algebraic integer if it is a zero of some monic polynomial in \(\mathbb{Z}[x]\). Show that the set of all algebraic integers forms a subring of \(\mathrm{C}\).

An element of \(\mathrm{C}\) is an algebralc integer if it is a zero of some monic polynomial in \(\mathbb{Z}[x]\). Show that the set of all algebraic integers forms a subring of \(\mathrm{C}\).