Example

(1) We shall define in no. 4 the field of rational numbers; In General Topology the field of real numbers will be defined (General Topology, IV, § 1, no. 3), also the field of complex numbers (General Topology, VIII, § 1, no. 1) and the field of quaternions (General Topology, VIII, § 1, no. 4). These fields are commutative with the exception of the field of quaternions.* (2) The ring $\mathbf{Z}/2\mathbf{Z}$ is obviously a field.

Let $K$ be a field. Every subring $L$ of $K$ which is a field is called a subfield of $K$ and $K$ is then called an extension field of $L$; it amounts to the same to say that $L$ is a subring of $K$ and that $x^{-1} \in L$ for every non-zero element $x$ of $L$. If $(L_i)_{i \in I}$ is a family of subfields of $K$, then $\bigcap_{i \in I} L_i$ is a subfield of $K$; for every subset $X$ of $K$ there thus exists a smallest subfield of $K$ containing $X$; it is said to be generated by $X$.