91Ó°ÊÓ

Find the splitting field \(K\) in \(C\) of the indicated polynomial \(f(x)\) over \(Q\), and determine \([K: \mathbb{Q}]\). $$ f(x)=x^{5}+x^{4}+x^{3}-x^{2}-x-1 $$

Let \(F \subseteq E\) be fields and let \(\alpha \in E\) be algebraic over \(F\). Show that $$ I=\\{f(x) \in F[x] \mid f(\alpha)=0\\} $$ is a proper ideal in \(F[x]\).

In Exercises 8 through 10 show that the indicated \(\alpha \in C\) is algebraic over \(Q\), and determine \(\operatorname{deg}_{\mathrm{O}}(\alpha) .\) $$ \sqrt{3}-i $$

Find the splitting field \(K\) in \(C\) of the indicated polynomial \(f(x)\) over \(Q\), and determine \([K: \mathbb{Q}]\). $$ f(x)=x^{3}+x+1 $$

Find a primitive element for the indicated field \(F\). $$ F=G F(9) $$

Find the splitting field \(K\) in \(C\) of the indicated polynomial \(f(x)\) over \(Q\), and determine \([K: \mathbb{Q}]\). $$ f(x)=x^{3}-3 x+2 $$

Find the splitting field \(K\) in \(C\) of the indicated polynomial \(f(x)\) over \(Q\), and determine \([K: \mathbb{Q}]\). $$ f(x)=x^{4}-4 x^{3}+6 x^{2}-4 x+1 $$

Find \(\operatorname{deg}_{F}(\sqrt{2}+\sqrt{3})\) for the indicated field \(F:\) (a) \(F=Q\) (b) \(F=Q(\sqrt{6})\) (c) \(F=Q(\sqrt{5})\) (d) \(F=Q(\sqrt{2}, \sqrt{3})\)

Find the splitting field \(K\) in \(C\) of the indicated polynomial \(f(x)\) over \(Q\), and determine \([K: \mathbb{Q}]\). $$ f(x)=x^{4}-2 x^{3}-x+2 $$

Find the minimal polynomial of \((\sqrt{2}+\sqrt{2} i) / 2\) over the indicated field \(F\) : (a) \(F=Q\) (b) \(\mathbb{R}\) (c) \(Q(i)\)