Exercise 4.3.3

Proof.

(a)

$(1,\sqrt[4]{2},{\sqrt[4]{2}}^2,{\sqrt[4]{2}}^3)$ is a basis of $\mathbb{Q}(\sqrt[4]{2})$ over $\mathbb{Q}$, and $(1,i)$ a basis of $\mathbb{Q}(i,\sqrt[4]{2})$ over $\mathbb{Q}(\sqrt[4]{2})$, thus $$(1,\sqrt[4]{2},{\sqrt[4]{2}}^2,{\sqrt[4]{2}}^3,i,i\sqrt[4]{2},i{\sqrt[4]{2}}^2,i{\sqrt[4]{2}}^3)$$

is a basis of $\mathbb{Q}(i,\sqrt[4]{2})$ over $\mathbb{Q}$

(b)

$(1,\sqrt[3]{2},{\sqrt[3]{2}}^2)$ is a basis of $\mathbb{Q}(\sqrt[3]{2})$ over $\mathbb{Q}$, and $(1,\sqrt{3})$ a basis of $\mathbb{Q}(\sqrt{3},\sqrt[3]{2})$ over $\mathbb{Q}(\sqrt[3]{2})$, thus $$(1,\sqrt[3]{2},{\sqrt[3]{2}}^2,\sqrt{3},\sqrt{3}\sqrt[3]{2},\sqrt{3}{\sqrt[3]{2}}^2)$$

is a basis of $\mathbb{Q}(\sqrt{3},\sqrt[3]{2})$ over $\mathbb{Q}$.

(c)

The minimal polynomial of $\sqrt{2+\sqrt{2}}$ over $\mathbb{Q}$ being of degree 4, $$\left(1,\sqrt{2+\sqrt{2}},{\sqrt{2+\sqrt{2}}}^2=2+\sqrt{2},{\sqrt{2+\sqrt{2}}}^3=(2+\sqrt{2})\sqrt{2+\sqrt{2}}\right)$$

is a basis of $\mathbb{Q}\left(\sqrt{2+\sqrt{2}}\right)$ over $\mathbb{Q}$.

(d)

w A basis of $\mathbb{Q}\left(i,\sqrt{2+\sqrt{2}}\right)∕\mathbb{Q}\left(\sqrt{2+\sqrt{2}}\right)$ being $(1,i)$, $$(1,\alpha ,{\alpha }^2,{\alpha }^3,i,\mathit{i\alpha },i{\alpha }^2,i{\alpha }^3),\mathrm{where}\alpha =\sqrt{2+\sqrt{2}},$$

is a basis of $\mathbb{Q}\left(i,\sqrt{2+\sqrt{2}}\right)$ over $\mathbb{Q}$.