number field

Definition 1.

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

Example 1.

The field of rational numbers $\mathbb{Q}$ is a number field.

Example 2.

Let $K=\mathbb{Q}(\sqrt{d})$, where $d\ne 1$ 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 $[K:\mathbb{Q}]=2$. We can explictly describe all elements of $K$ as follows:

$$K=\{t+s\sqrt{d}:t,s\in \mathbb{Q}\}.$$

Definition 2.

A number field $K$ such that the degree of the extension $K\mathrm{/}\mathrm{Q}$ is $\mathrm{2}$ is called a quadratic number field.

In fact, if $K$ is a quadratic number field, then it is easy to show that $K$ is one of the fields described in Example $2$.

Example 3.

Let $K_n=\mathbb{Q}({\zeta }_n)$ be a cyclotomic extension of $\mathbb{Q}$, where ${\zeta }_n$ is a primitive $n$th root of unity. Then $K$ is a number field and

$$[K:\mathbb{Q}]=\phi (n)$$

where $\phi (n)$ is the Euler phi function. In particular, $\phi (3)=2$, therefore $K_3$ is a quadratic number field (in fact $K_3=\mathbb{Q}(\sqrt{-3})$). We can explicitly describe all elements of $K$ as follows:

$$K_n=\{q_0+q_1{\zeta }_n+q_2{\zeta }_n^2+\mathrm{\dots }+q_{n-1}{\zeta }_n^{n-1}:q_i\in \mathbb{Q}\}.$$

In fact, one can do better. Every element of $K_n$ can be uniquely expressed as a rational combination of the $\phi (n)$ elements .

Example 4.

Let $K$ be a number field. Then any subfield $L$ with $\mathbb{Q}\subseteq L\subseteq K$ is also a number field. For example, let $p$ be a prime number and let $F=\mathbb{Q}({\zeta }_p)$ be a cyclotomic extension of $\mathbb{Q}$, 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:

$$F^{+}=\mathbb{Q}({\zeta }_p+{\zeta }_p^{-1}).$$