Relation between characterstic and cardinality of a field

Question

I am studying Galois Theory from Hungerford's book and while studying lemma on page 280 ( lemma 5.5).

Does there exists a field which is not finite but characterstic is finite?

This question arised in my mind as I was reading lemma5.5 as here if F is finite, then $\phi$ is an automorohism otherwise monomorphism but in both cases it has cardinality p.

I think there doesn't exist ( if isomorphism of elements are considered)but why as there would be only {0,1,..., p-1} elements(distinct) due to characterstic being p, but isomorphisms of elements are not concerened then I thing there will exist.

So, should isomorphism be considered?

Answer 1

Take the algebraic closure of a finite field.

Any field has an algebraic closure, on one hand. On the other, any algebraically closed field is infinite.

This gives a whole host of examples.

Answer 2

Take $\mathbb{F}_{p} (t) $ denoted for all rational functions with coefficients in the field $\mathbb{F}_{p}$ and $p$ is prime.