The characteristic of a field F is defined as either:
a) The number of times one must add the multiplicative identity of F to get the additive identity of F. If the additive identity is never reached the field has characteristic zero.
b) The prime subfield P of F is the intersection of all subfields of F. F has characteristic p if P is isomorphic to $\mathbb{Z}/p\mathbb{Z}$ or characteristic 0 if it is isomorphic to $\mathbb{Q}$.
Simply, why are these definitions equivalent?