Prime Field

Description

The prime field is something like the skeleton of a field. It is the smallest subfield. This is something like the counterpart of the Algebraischer Abschluss which is the greatest extension field.

Definition

Let $K$ be a field and $F_i$ be all subfields. Then $P(K)={\bigcap }_{i\in I}{F}_i$ is the prime field.

Properties

1. If $char(K)=0$, then $P\cong \mathbb{Q}$
2. If $char(K)=p$, then $P\cong {\mathbb{F}}_p$

$char$ bezieht sich auf die Charakteristik (Ring).