Does there exist an infinite field with characteristic $p$ for any prime $p$ that is not too big?

Question

Does there exist an infinite field with characteristic $p$ for any prime $p$?

I haven't seen any such fields in my course yet.

Answer 1

take the field of fractions of polynomials in $F_p[x]$, i.e. the set of elements

$P(x)/Q(x)$

where $P$, $Q \in F_p[x]$ (and $Q \ne 0$).

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The answer linked by mt is by far more complete than mine!