Field_of_Prime_Characteristic_has_Unique_Prime_Subfield

Let $F$ be a field whose characteristic is $p$. Then there exists a unique $P \subseteq F$ such that $(1): \quad P$ is a subfield of $F$ $(2): \quad P \cong \Z_p$. That is, $P \cong \Z_p$ is a unique minimal subfield of $F$, and all other subfields of $F$ contain $P$. This field $P$ is called the prime subfield of $F$.