# Characteristic

For a field F, the characteristic is the smallest natural number n such that 1F+...+1F n times yields 0. If no such n exists, we say that the characteristic is 0.

Examples and notes:

```* For any ordered field (for example, the rationals) the characteristic is 0
* For any Z/p (the field of integers module a prime number), the characteristic is p
* For any field, the characteristic is either 0 or prime.
* Any field contains a copy of exactly one member of {Z/p}U{Q}. If it contains Q, its characteristic is 0 and otherwise its characteristic is p for the unique Z/p it contains
* As a corollary, if one fields contains another, their characteristic is the same. In particular Z/p[x]/q(x) for a prime polynomial q(x) over Z/p has characteristic p
* As another corollary, any field of 0 characteristic is infinite.
* As another corollary, the real and complex numbers also have 0 characteristic
* However, there are infinite fields of prime characteristic. For example, the field of all rational functions over Z/p is one such. The algebraic closure of Z/p is another one such.
* The size of any finite field of characteristic p is a power of p. Since in that case it must contain Z/p it must also be a vector space over Z/p and from linear algebra we know that the sizes of finite vector spaces over finite fields are a power of the size of the field.
* This also shows that the size of any finite vector space is a prime power. (It is a vector space over a finite field, which we have shown to be of size pn. So its size is (pn)m = pnm. QED)
```