Definition 1   A field which is a finite extension of $\mathbb{Q}$ , the rational numbers, is called a number field (sometimes called algebraic number field). If the degree of the extension $K/\Rats$ is $n$ then we say that $K$ is a number field of degree $n$ (over $\Rats$ ).

Example 1   The field of rational numbers $\Rats$ is a number field.

Example 2   Let $K=\Rats(\sqrt{d})$ , where $d\neq 1$ is a square-free non-zero integer and $\sqrt{d}$ stands for any of the roots of $x^2-d=0$ (note that if $\sqrt{d}\in K$ then $-\sqrt{d}\in K$ as well). Then $K$ is a number field and $[K:\Rats]=2$ . We can explictly describe all elements of $K$ as follows: $$K=\{ t+s\sqrt{d} : t,s \in \Rats \}.$$

Definition 2   A number field $K$ such that the degree of the extension $K/\Rats$ is $2$ is called a quadratic number field.

Example 3   Let $K_n=\Rats(\zeta_n)$ be a cyclotomic extension of $\Rats$ , where $\zeta_n$ is a primitive $n$ th root of unity. Then $K$ is a number field and $$[K:\Rats]=\varphi(n)$$ where $\varphi(n)$ is the Euler phi function. In particular, $\varphi(3)=2$ , therefore $K_3$ is a quadratic number field (in fact $K_3=\Rats(\sqrt{-3})$ ). We can explicitly describe all elements of $K$ as follows: $$K_n=\{ q_0+q_1\zeta_n+q_2\zeta_n^2+\ldots+q_{n-1}\zeta_n^{n-1} : q_i\in \Rats \}.$$ In fact, one can do better. Every element of $K_n$ can be uniquely expressed as a rational combination of the $\varphi(n)$ elements $\{\zeta_n^a : \gcd(a,n)=1,\ 1\leq a < n\}$ .

Example 4   Let $K$ be a number field. Then any subfield $L$ with $\Rats \subseteq L \subseteq K$ is also a number field. For example, let $p$ be a prime number and let $F=\Rats(\zeta_p)$ be a cyclotomic extension of $\Rats$ , where $\zeta_p$ is a primitive $p$ th root of unity. Let $F^+$ be the maximal real subfield of $F$ . $F^{+}$ is a number field and it can be shown that: $$F^+=\Rats(\zeta_p+\zeta_p^{-1}).$$