Section 7.4
Fields

Let K be a field. Every subfield of K contains 0 and 1, and so it also contains

1 + ··· + 1     and     -1 -1 - ··· -1.

The subfield therefore contains all integral multiples of 1 and -1 as well as all fractions of these multiples (as long as the denominator is nonzero). These elements make up a subfield themselves.

Theorem

A field generated by the empty set (or by 0 and 1), is isomorphic with Q or Z/pZ for some prime number p.

In particular, every field contains a subfield isomorphic with Q or Z/pZ for some prime number p.


We consider the smallest subfield L of K (it is generated by 0 and 1).

Definition

If L is isomorphic with Q, then K is said to have characteristic 0.

If L is isomorphic with Z/pZ, then K is said to have characteristic p.

By the above theorem, the characteristic of a field is either zero or a prime number.