Section 1
A
Find a basis for $\mathbb{Q}(i \sqrt{2})$ over $\mathbb{Q}$, and describe the elements of $\mathbb{Q}(i \sqrt{2})$. (See the two examples immediately following Theorem 1.)
Show that every element of $\mathbb{R}(2+3 i)$ can be written as $a+b i$, where $a, b \in \mathbb{R}$. Conclude that $\mathbb{R}(2+3 i)=\mathbb{C}$.
If $a=\sqrt{1}+\sqrt[3]{2}$, show that $\left\{1,2^{1 / 3}, 2^{2 / 3}, a, 2^{1 / 3} a, 2^{2 / 3} a\right\}$ is a basis of $\mathbb{Q}(a)$ over $\mathbb{Q}$. Describe the elements of $Q(a)$.
Find a basis of $\mathbb{Q}(\sqrt{2}+\sqrt[3]{4})$ over $\mathbb{Q}$, and describe the elements of $\mathbb{Q}(\sqrt{2}+\sqrt[3]{4})$.
Find a basis of $\mathbb{Q}(\sqrt{5}, \sqrt{7})$ over $\mathbb{Q}$, and describe the elements of $\mathbb{Q}(\sqrt{5}, \sqrt{7})$. (See the example at the end of this chapter.)
Find a basis of $\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})$ over $\mathbb{Q}$, and describe the elements of $\mathbb{Q}(\sqrt{2}, \sqrt{3},$, $\sqrt{5}$.
Name a finite extension of $\mathbb{Q}$ over which $\pi$ is algebraic of degree 3