91Ó°ÊÓ

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

In Exercises 10 through 17 determine whether the indicated subset \(U\) is a subspace of the indicated vector space \(V\) over the indicated field \(F\). $$ U=\\{f(x) \in Q[x] \mid f(1)=0\\} \quad V=Q[x] \quad F=\mathbb{Q} $$

In Exercises 13 through 17 find a basis for the indicated extension field of \(Q\) over \(Q\). $$ \mathrm{Q}(\sqrt{2}, \sqrt{3}) $$

In Exercises 10 through 17 determine whether the indicated subset \(U\) is a subspace of the indicated vector space \(V\) over the indicated field \(F\). $$ U=\\{f(x) \in Q[x] \mid f(1)=2\\} \quad V=Q[x] \quad F=Q $$

In Exercises 13 through 17 find a basis for the indicated extension field of \(Q\) over \(Q\). $$ Q(\sqrt{2}, i) $$

In Exercises 10 through 17 determine whether the indicated subset \(U\) is a subspace of the indicated vector space \(V\) over the indicated field \(F\). $$ U=\\{(x, 0, z) \mid x, z \in \mathbb{R}\\} \quad V=\mathbb{R}^{3} \quad F=\mathbb{R} $$

In Exercises 10 through 17 determine whether the indicated subset \(U\) is a subspace of the indicated vector space \(V\) over the indicated field \(F\). $$ U=\\{(x, 1, z) \mid x, z \in \mathbb{R}\\} \quad V=\mathbb{R}^{3} \quad F=\mathbb{R} $$

Show that the number of primitive \(n\) th roots of unity is \(\phi(n)\).

In Exercises 13 through 17 find a basis for the indicated extension field of \(Q\) over \(Q\). $$ \mathrm{Q}(\sqrt{2} i) $$

Construct the subfield lattice of the indicated field \(F\). $$ F=G F\left(2^{6}\right) $$