Algebraic number field

Remarks

Example: $K = \Q(i) = \set{a + bi : a, b \in \Q}$, a subfield of $\C$. It contains $\Q$, and $1, i$ is a basis over $\Q$, so $[K : \Q] = 2$ and $K$ is a number field. Non-example: $\R$ contains $\Q$, but it is not of finite dimension over $\Q$, since a finite-dimensional space over $\Q$ is countable and $\R$ is not. Another non-example: the field of all algebraic numbers is an algebraic extension of $\Q$ of infinite degree. Every element of a number field is an algebraic number, and by the primitive element theorem every number field is $\Q(\alpha)$ for one algebraic number $\alpha$. The elements of $K$ that are roots of a monic polynomial with integer coefficients form a subring $\mathcal{O}_K$, the ring of integers of $K$.